I hadn't really thought about it, but it could be that your Asian amp employs circuits that have the same familiar heritage as do many of the more popular American/British amps. It would be an informative project to draw out the circuit diagram just as @beans-on-toast recommended -- it may be more 'normal' than I had originally thought. There may well be a lot to recommend it as a mod test-bed for your own enlightenment, but I wouldn't pay a tech to mess with it. And I don't think the output transformer is necessarily a place to start -- I doubt that's the limiting factor in the amp's performance. Nor is the 12AX7 preamp tube, IMO. But as I say, it might be a decent vehicle to start learning some more about how tube amps work.
Could be a clone of the Laney CUB 12R according this thread on this forum:
Monoprice 15 watt tube amp problems - diyAudio
But I don't give any guarantee it is.
 
...the amp has a PCB and surface mounted devices....
I guess that kinda puts the kibosh on using it as any sort of mod test-bed, unless you want to discard the PCB and rebuild it with perf board or something. Actually, for $100, you got a power supply, a cab and speaker, and a chassis with tube sockets and tubes already on it. Plus (maybe) some jacks and pots. That's really not a bad place to start if you wanted to build your own amp.
 
Here’s something to consider; I have two rigs I play both guitar (twangy, slightly gritty) and bass (fingerstyle rock P-bass on flatwound strings) through. I have noticed that for my ear and playing style, the settings on the amp controls don’t change much between instruments! I might make small tweaks on the lows or a slightly different low-mid cut between bass and guitar, but I could leave it one way and be in the final ballpark. EQ, gain settings...

I happen to have speakers and enclosures that sound good to me with both, AND THAT IS THE KEY. My preamp is basically doing a bright almost-clean sound with a little boost in the upper mids and a little cut in the lower mids. No special tube or gain magic going on. I like a little compression but it’s not a must-have.

So I don’t think you should dump money into modding the circuit. You’ll be able to work with that...if you are feeding a well-designed box designed for a good bass driver. The speaker and enclosure are where the fun is.

Since you won’t be bass gigging with a 15 watt tube amp, why not consider some other speaker sizes? Maybe a little 1 x 8 cab would be fun for home or coffeeshop or basement acoustic jams? You could get away with a smaller box that way. The four FaitalPRO 8s in my AudioKinesis cab sound great with four-string bass, can reportedly handle five-string too, and their frequency response extends high enough to reproduce guitar and lap steel pretty well too...in a dual-ported 14 x 14 x 18 box.

Those are expensive little drivers, but there are others to check out...the US Speaker site can swallow you for hours, reading descriptions and rubbing your hands together.
 
I guess that kinda puts the kibosh on using it as any sort of mod test-bed, unless you want to discard the PCB and rebuild it with perf board or something. Actually, for $100, you got a power supply, a cab and speaker, and a chassis with tube sockets and tubes already on it. Plus (maybe) some jacks and pots. That's really not a bad place to start if you wanted to build your own amp.
Here I thought all amps, except those advertised as point to point, are made with PCB anymore. That is a cool idea though, especially seeing the how to’s on robrob’s page. I will admit, however, the numerous warnings about how an amp can kill you if you touch it wrong have me hesitant to take a crack at it myself.
 
Here I thought all amps, except those advertised as point to point, are made with PCB anymore. That is a cool idea though, especially seeing the how to’s on robrob’s page. I will admit, however, the numerous warnings about how an amp can kill you if you touch it wrong have me hesitant to take a crack at it myself.
There is a difference between PCB and through-hole versus SMT (both are forms of PCB parts/assembly methods). It has NOTHING to do with point to point.
 
There is a difference between PCB and through-hole versus SMT (both are forms of PCB parts/assembly methods). It has NOTHING to do with point to point.
I’m saying that if it’s not point to point I wouldn’t have any clue how to work on it myself. Even then, I wouldn’t know what I’m doing, I’d just know how to properly solder the joints, which is more than I can say about working on a PCB of any kind, through-hole or otherwise.
 
I would definitely avoid closing the back on the existing cabinet. Low priced guitar combos with EL84 valves hanging downwards from PCB mounted sockets often have issues anyway, and that's without shutting off all ventilation.