EL34 in triode vs 300B/845 SET

An amplifier that operates push-pull with 300bs is likely not in the budget as proposed. I suspect that the amplifier that will be used will be an ultralinear amp based on 6550s/KT88s.

Only few people may agree but I firmly believe good sound comes from high current not high watts even though inter-related to some degree.

@hifineubeeThere really isn't any such thing as 'high current'; that's a bit of story created by solid state amp manufacturers. Current does not exist without voltage, and together the two are power. Its a mathematical relationship!

The reason 'neutral' (usually solid state, as opposed to 'warm') might seem unpleasant is due to how the amp distorts. In the case of solid state, the distortion seems minimal, but to the ear it is not, as the distortion in question is all higher ordered harmonics, to which the ear is keenly sensitive (the ear uses these harmonics to sense sound pressure, so has to be able to deal with 140db range, hence the sensitivity). So 0.0x% numbers in solid state amps are easily audible, heard as brightness and harshness.
Tube amps generally don't make as much higher ordered harmonics and so sound smoother and more detailed. Tubes usually do make more distortion overall, but its mostly lower orders and the ear simply is less sensitive to the lower orders. In a way, you can argue that tubes more closely adhere to the rules of human hearing perception. In a nutshell.
I'd be looking at something with 60 watts or so. You can find amps of that power inside of your budget, and that is significantly more power (and likely lower distortion and wider bandwidth too) than what you are running right now- you'll notice the improvement instantly!

- I've started pairing up my 300B SET with a subwoofer to help with 40hz and below base, I'm wondering if this is as good as it gets so I should just save my money and not further go down on another tube amp.

One of the prices you pay with higher power SETs is that they might play bass and they might play highs, but not both at the same time. The bigger the amp is, the more this is the case.

- Do bigger and heavier tube amp transformer directly improve sound quality? Should I entertain more expensive 300B amps with heavier transformers like those from Sophia Electric?

Some transformers are potted, and so appear larger and heavier than they really are.

Seriously, if you plan to keep your speakers, you'll have a lot more luck with an amp of considerably more power- and it won't be SET.

The issue here appears to be that even though this system appears to work, as Trelja points out, neither the amp or speaker is being shown in its best light.
IOW, it can get a whole lot better, but to do so you either need a lot more power (not available from an SET) or get a speaker that is a good 10-12 db more efficient! This despite how good it seems to sound right now; but the OP is correct in his surmise that more power (or more efficiency) will help things out.

The problem is that SETs have bandwidth limits that get more pronounced with the greater power they can make. That is why the lower power units sound better.
BTW, there was one comment about push-pull having crossover distortion; this is not true in most cases. Crossover distortion is an artifact of class B operation, and to my knowledge no home audio gear employs class B, so that's a bit of a red herring. It is the most common complaint against push-pull, but as best I can make out, is mostly borne out of not understanding push-pull technology all that well.