Right: the tl;dr is that the nature of Rachni holdings means that they can't really sustain keeping their entire navy on constant, active status guarding relays if they want to do anything else. This presents a vulnerability for you in bashing open a relay for long enough to slip raiding forces past the initial defensive line without having to actually face their full, on-paper strength. Once on the secondary network, raiders can basically keep jumping until they find a target they like. Getting fleets back is the tricky part, but less so than you'd think. You win even if you burn nothing but the Rachni are forced to mobilize the forces it would take to actually stop you, given the cost involved. There exist ways to counter this doctrine, but given that trans-relay commerce and logistics raiding has never been a thing, those tactics don't exist, giving you the breathing room to keep ahead of innovations with your own ideas. I do not give options that don't work due to the way the setting runs.
The long version:
The most obvious barrier to a viable raiding doctrine is relay defense. Any opponent with a brain will post defense forces to relays, and raiding forces are not built with pitched battle in mind -- quite the opposite, in fact. Conventional wisdom states that no force can advance beyond the relay without securing it -- after all, to do so would leave a hostile force at their back. Utter madness. The proof of this is in your own experiences over the course of your first term -- the Rachni never did opt to circumvent your forces at the relay. They could not; even Rachni do not throw away ships for nothing.
The brainwave Beshkar had was rooted in what happened in the Battle of Hercules. Namely, even when the Rachni were preparing for an imminent assault, even when sending a truly formidable reconnaissance-in-force, even when tensions were the highest they had ever been...
...a huge portion of their fleet was still in dock, well away from the relay.
This doesn't appear to make sense. Virmire can keep its fleets fully mobilized at the relay at all times. Why would the Rachni not, particularly when tension was so high and conflict was so obviously imminent?
The answer to that is a little less immediately apparent, but it's obvious in hindsight. It is true that the Rachni control massive swathes of the galaxy. However, they have not penetrated any of the Citadel's core space; the territory they have taken is colonial space, nearly bare of heavy industry. Whatever they have, they will need to have constructed themselves while paying the immense costs of an attrition-based naval doctrine against the entire galaxy. Given the state of Hercules -- scans of Eletania, a near-garden world if it weren't for the allergen issue, about which Rachni would not care, indicate precious little extensive development -- it is clear that they have not been able to build much.
From this, Beshkar concludes that while the Rachni may or may not run extraction operations within their controlled space, the bulk of their production centers are back in their home systems. Thus, basically the entire rest of their space is composed of supply line.
This is significant. Virmire can field its fleets on permanent active status because it does so within a thirty minute jog of a heavily-industrialized garden world. If a ship is damaged or needs refuel and maintenance, it can turn around and fly home, and be back in the fight soonest. Even just moving to Hercules, establishing local ship basing has already proven to be valuable -- relay travel is not stressless, and damaged vessels are much better served by local shipyards than further sharp jolts before a safe berth. The Rachni, however, must be shipping parts across multiple jumps and thousands of light years. Doing so is not trivial even with relays. Supply ships break down and burn fuel, and accidents happen to shipments. With the supply line so much less robust, then, it is simple, financial sense to keep ships in dock rather than incurring the cost of keeping all of them ready to fight on a moment's notice.
Therefore, when Virmire's regular fleets jump through a relay to clear a path for raiding forces, they will face a far more limited array of forces than the Rachni's on-paper strength would suggest. The key to Beshkarian Doctrine is in exploiting this vulnerability. If Virmirean forces were interested in securing a system, they would need to face the entire strength of the enemy fleet one way or the other. Under the raiding doctrine, they are not so interested. They are interested in sweeping aside the tripwire forces at the relay, calling through the raiding fleet, and then firing salvos at local ship basing and planets, forcing hostiles to screen with their own ships for hours or days as the rounds crawl through space. Meanwhile, the raiders jump to the relay they wish to use and then flash away, behind the main defenses and able to run roughshod over rear echelon garrisons. Once your raiders are on the secondary relay network, they can just keep jumping until they find a juicy and vulnerable target.
You may ask: how do we get these forces back? Well, that's a matter of careful scheduling; the raiders get a timetable of [insert duration here] to return to the same system, whereupon your forces launch an assault to make room. This is the difficult part, but with the raiding forces coming in from the flanks, you actually have a decent position.
Raiding doctrine has two objectives, either one of which constitutes a successful campaign. The first is to wreck strategically-significant amounts of hostile infrastructure, limiting their ability to resupply their forces away from their industrial heartlands and enabling conventional offensives of your own. This is the ideal. However, an acceptable secondary outcome is to force the enemy to deploy massive fleet elements all throughout their space in an effort to flush out or contain your raiders. Ordinarily, this would be relatively simple for the Rachni; a strong cruiser force should be able to hunt down any raiding fleet in relative security, in conjunction with defense platforms. However, you have battlecruisers. In order to address battlecruisers, the Rachni need to position massive fleets of cruisers or outright dreadnoughts in good response positions on active status. Neither of these is a cheap option. Neither of these is a reliable option. Either results in the Rachni's ability to initiate offensives against you being critically hampered as much of their strength is tied up elsewhere, consuming oceans of supplies to keep running as they police their own space.
Now, I don't need to tell you that there exist tactics to counter this doctrine. However, the Rachni don't have these tactics. They will begin developing them once or if they encounter a need -- and there has never been a need for trans-relay counter-raiding tactics before. Only in contested clusters does commerce and logistics raiding occur. You will have a relatively significant period of time while the Rachni reconfigure parts (but not all) of their massive war machine to counter your brand-new innovation. By that time, you will hopefully be taking options to innovate your doctrine yourself (or your High Admiralty will be making good passive rolls to institute doctrinal reforms), countering their counters.
Raiding doctrine has obvious weaknesses. If it's unsuccessful, it leaves you facing down a pissed-off enemy with forces that pack a bigger punch in a fleet battle. There is the potential, if disaster strikes, to lose whole fleets in enemy territory and never know until you launch a futile assault to cover a retreat that will never happen. Other doctrines have their own weaknesses that can spell disaster.
However, you may rest assured that the weaknesses of a given doctrine -- and raiding doctrine in particular -- do not rest in its soundness as a concept. For heaven's sake, do you really think I'm going to feed you options that can't hope to achieve what they set out to do? I am Lawful Evil! If nothing else -- if you pick the worst possible doctrine, one which will critically fail to address your strategic situation -- it will at least be able to do the thing that's in its name, and not be hamstrung by the very premises of the setting. By enemy action, possibly. But not because I'm giving you a blatantly unsound option.