I haven't done that exact method before. I've cooked a custard/queso blanco hybrid before, though.
It's challenging because your timing has to be impeccable. As you said, mixing it back in is tough. You are right, extra solids will affect how the micelles bond and your curd set will be weaker. But, if you wait, you give rennet a chance to work. Rennet doesn't like a lot of extra salts and solids. Basically, what I'm trying to say is you wait for 50% of the casein micelles to be cleaved, then add whatever you want, stir quickly, and wait. Disturbing the milk is actually not a huge deal unless you agitate it violently. if it's moving slightly, that's ok. At flocculation, it will suddenly stop moving.
If you're making a processed cheese, the key consideration is that it will not "age". You have to start out with the final flavor that you want. The classic beginning of processed cheese happened when an aged swiss style was melted, sodium citrate was added, and the whole mass was left to solidify into a block. Nowadays, it's similar. The principle is you melt, add an emulsifier, and augment with additives to get the fat and protein and moisture numbers right.
So if you want a fresh cheese taste that's still creamy and seasoned like egg nog, you could augment pH with citric acid, but remember citric has a different flavor profile than lactic. Also, the bacteria do add a little flavor. You can do whatever process you want, but flavor will be different depending on the variables. Also, be careful of the moisture content. You want regular curds, not too moist, not too dry. You will need to heat the curds so they are liquified and then add in citrate, so they do need to melt... not necessarily stretch.
By citrate I meant sodium citrate. I think these additives are technically called emulsifying agents. Some other ones are Rochelle salt (sodium or potassium tartarate) and sodium metaphosphate. But really, sodium citrate is the standard one.
Farmer, I read a lot... but there are degrees in food science that teach all these details, usually at the masters+ level. Will reply to PM.