Honestly, my hope for if we do go for a the Doofenshmirtz Space Program would be that we roll well enough (or do well enough) to hit the milestone of "first commercial launch provider with semi-reusable launch vehicles". Even OTL, the tech theoretically exists to make this happen already in this timeline - it's just a matter of someone putting it all together.
My plan for a launch site would actually be to see if we can convince Shego to play nice, and let us use a launch complex in Texas, on the coast of the Gulf of Mexico. That allows us to keep our personnel, rockets, and equipment as
far the fuck away from DOR-15 as practical, while also giving us access to more or less every orbit that Cape Canaveral would have - although, admittedly, we might need to negotiate with Mexico for some of the lower latitude possibilities for launch sites on the Texas coast. That being said, if we do go for Texas, another possible up-side is the fact that, if we do go for a low latitude coastal launch site, and manage to negotiate successfully with the Mexican government, we might be able to get away with not needing anything like SpaceX's drone ships - for launches where RTLS (return to launch site) isn't an option, they could instead touch down in Yucatan.
If we wanted to go a different route from "pulling a SpaceX" with VTVL reusable first stages and expendable second stages, my vote would be to see if we could do something like the
Space Transportation System from Polish Eagle's
Right Side Up: A History of the Space Transportation System timeline - which is basically a winged
S-IC (as in, Saturn V first stage) with an upgraded (and mass-production ready)
S-IVB stage acting as an expendable upper stage, with crew delivery handled by a small reusable space-plane able to carry along light cargo (IIRC about 8 tons) for satellite servicing missions or scientific observations, with that space-plane being small enough to have useful and existent abort modes for the entire duration of it's flight. The STS' first stage (in our version) would just be unmanned instead of needing it's own flight deck/abort capsule, and would probably also use (as it's air-breathing engines) something a bit more advanced than the
General Electric F110, just due to being built in the 21st century if nothing else. Possibly the
Pratt & Whitney F135, if we can convince the government to let us buy them, due to both being more advanced than the F110, and also producing more thrust even without an afterburner - thus potentially allowing for the usage of fewer engines, thus reducing weight and fuel consumption during aerodynamic flight.
As for rocket motors, a somewhat obvious choice (if any development work was done on them ITTL) would be the
F-1B engines proposed for use as booster engines on the SLS - they're cheaper and simpler to make than F-1 or F-1A engines, while producing almost as much thrust as the F-1A. They would need to be modified for in-flight restarting, to allow for boost-back/deceleration burns (to prevent the stage from simply melting or needing highly complicated thermal protection systems), but would allow DEI's Evil Launch Services Subsidiary (EvilspaceX?) to not have to do as much technical development work. Alternatively, we could develop a new engine - 4,450 kN (kilonewton) thrust class engine would allow a cluster of 9 of them to match a cluster of 5 of the F-1B, while a 5,720 kN thrust class engine could be used in a cluster of 7 to replace a S-IC stage's main engines. For second stages, one potential option would be something like the J-2X - it's got more thrust and specific impulse than the J-2S, and while it technically has less performance than the "J-2S-2" used in the Right Side Up timeline, it's only by about 3 seconds, which can be easily made up with simply making the second stages out of modern materials and with modern methods to reduce dry weight (as well as using modern computers, which are
much lighter than their mid-late 1970s counterparts).
As for the Orbiter, the only change from the Right Side Up timeline version (other than figuring out a reusable TPS other than a ceramic tile solution) I can think of would be designing something other than the LR91 for use as launch abort motors - probably something vaguely akin to the
SuperDraco engines developed by SpaceX in our timeline. By using eight pairs, we could have enough thrust for a useful "zero-speed" launch abort - ie., something goes wrong on the pad, and the crew just use the abort motors on the space-plane to get clear instead of a risky "zip-line" escape as used on the OTL!Space Shuttle for launch pad aborts. If we fitted, say, the inner most pairs with extendable nozzles for use in vacuum, that would also give us some
very useful OMS engines, which could burn the same MMH (monomethyl hydrazine) and NTO (dinitrogen tetroxide) propellant as the space-plane's RCS (reaction control system).
Of course, all this is moot if we somehow wound up with a crit and managed to get some sort of SSTO, or a space tether based solution, or any number of other things.