@digiwolf88
If stopped, you need all the vehicle paperwork including a registration document (and suitable supporting docs if not in your name), current inspection (not needed if new), and valid insurance. You should also have a vignette in Bulgaria (and a couple of other countries you drive through to get here).
You don't need to register as soon as you arrive. There is some grace period (months) for foreign-registered cars. However, the dealer might be giving you some kind of temporary plates, so you'll be subject to a tighter deadline.
There should be no issues driving a German-registered vehicle to and into Bulgaria (subject to your documentation above). Customs/immigration are typically not concerned with where your vehicle is registered. This is done at your local KAT.
There is some good info at BulgarianPlates (dot com), and I (and a few others here) have used them. They mostly use Plovdiv KAT, but I doubt that's an issue for an EOOD.
I don't know anything about VAT. I suspect it's a wash, and you either pay it there or here. My guess is that you should probably tell the dealer that it's for export to elsewhere in the EU, and they won't charge VAT. But then you'll have to declare and pay it here. Personally, I'd buy a recent model used car with low mileage as they've already incurred significant depreciation, and VAT isn't applicable. But I'm a cheapskate. :-) Maybe this link will help:
https://europa.eu/youreurope/citizens/v … dex_en.htm
KAT is not difficult, it's just a tedious day waiting in line for different parts of the registration process. I'd recommend that you pay an attorney (or suitable local helper) to handle it for you, especially with the extra paperwork for a company vehicle. In this case, you can be there too... or you can outsource it completely by giving them a POA and then they just call you when it's done. Of course, you can't give a POA to just any random helper who says he can sort it all out for you. :-) The minimum you need is an interpreter. My local state-registered guy charges 50 lv an hour, and you might well spend most of the day there (depending on when you arrive, and how busy it is).
Separately, you might consider if it's better to deal with your personal residence here (if you haven't already). As an EU citizen it's pretty easy, and the EU Citizen Registration can be done in a couple of days, and you get a biometric residence permit. In this case, you can own the car in your personal name if you want to.