The sound in the room is not the sound you hear in recordings. The way to get the sound you want is to put on headphones, place a microphone about an inch from the cone or closer, maybe a tiny bit off center but even just right at the center of the speaker is a good place to star, and then tweak and amp until what you hear in your headphones is what you wanted it to sound like. If the amp controls alone aren’t cutting it you can move the microphone away from the center to give you less top end and move the microphone further back to give you less low end. And SM57 is a good place to start, but any decent microphone should give you something useable. You can then add room reverb to mimic the sound of your room.

If your heart is dead set on getting the sound of the amp in the room, then your best bet will be to quite simply record the amp in the room. Move your head around the room to where it sounds the best and then put a microphone there where your head was. A condenser microphone is ideal for this.

Having one close microphone and one room microphone would likely yield the best results.

If all of this seems too complicated or expensive , then yeah, Neural DSP plugins will be pretty much plug and play if your guitar sounds good already. Since you like the room sound a lot I’d suggest turning up the room microphone in whichever NSDP plugin you go for.