Honestly whether they continue now depends entirely on what else, if anything, is still en route from Norn, or what professionals they held in reserve. I'm assuming a worst case scenario of another dwarven infantry regiment or tactical equivalent that we'll need to aggressively dislodge before we can call this campaign over, but I'll be happy to be proven wrong.
Another Dwarven regiment isn't going to save them. That would only push their force to four reduced cavalry regiments and two infantry regiments. That's not viable if it's facing anything bigger than a division, which they are.
Keep in mind that the Dwarven regiments that are reinforcing them aren't Arnése royalists. They are Nornisch regulars. While they're working together, they are different factions with different recruiting bases. The loss of two regiments of regulars, maybe three if they come around for another go or had we waited, hurts. But they can be fundamentally replaced by drafting more people into the army and raising new regiments.
The royalists cavalry is irreplaceable and their infantry is also coming from a very limited recruiting base. Without holding any ground in Arné they have just whoever is willing to flee the country to recruit from. A big ask for (former) peasants that form their main recruitment demographic. Their officer base is even more limited given it is restricted to actual (not titular) nobility.
To make a bit of an example let's look at the émigre armies. The Volunteer Army was 8100 men strong, with 2000 of those being Norns. Had we withdrawn to Vaud chances are it would have been 9100 with 3000 Norns.
OTL we had the following émigre armies (not émigres who joined armies of other nations, but independent formations). The Army of the Princes and the Army of Condé.
The Army of the Princes was short lived. Lasting only from 1791 to 1792 and the first year in pretty much name only. It counted 10.000 men at its hight, mostly raised by Louis XVI's brothers in Trier. It mainly served in the rear of Brunswick's army, never actually taking the field. It was also severely distrusted by both the Austrians and Prussians, and considered unreliable in combat. It was hated by the Rhenish population.
The Army of Condé lasted longer, from 1791 to 1801, and actually saw action. It was noted as officer heavy, with half of the original intake being noble officers (about 2000) who fled the revolution. During the 1792 wars it was also held in reserve due to concerns about its reliability, even if it was 20.000 men strong at that point. After Valmy there were talks about disbanding it, but it was taken into Austrian and Russian service (with British pay) fighting in numerous battles and would fluctuate between 6.000 and 10.000 men for the remainder of its service.
Recruitment and funding was an issue in both cases, with replacements and steady financing being core issues. Eventually their remnants, especially the officers, would mostly be integrated into the national armies of the coalition members.
To throw in some contemporary assessments on their skills (google translated from French).
"The emigrants, individually very brave, are bad infantry, and this body should be supported and guided by a troop more accustomed to discipline and fatigue. Children, old people, office staff fill the ranks. They fall after a walk, they clutter hospitals. The captains who carry the musket know the price of their devotion, and demand that their comrades who serve as their leaders do not treat them like simple soldiers. »
"The legions composed of gentlemen, reduced to pay and soldiery, were destroyed by disease; if we except a few vigorously constituted individuals, all those who escaped death returned in a state of exhaustion and infirmity which they will feel all their lives. To form corps of gentlemen-soldiers would therefore be to complete the destruction of the remains of the French nobility, half of which have already perished. »
Again. Assuming some similarity in numbers, then the kind of losses the royalists are suffering here are not sustainable in the long run. I'm expecting their total manpowerpool as a relevant faction to be about 20.000 ~ 25.000 at maximum, probably lower.
