So now that I presented most of the problems, what should you do?
Feel free to raise/lower the impedance of the load attached to the secondary, aka yours speakers. With an 8Ω output tap, add a second 8Ω speaker. Put it in parallel for a 4Ω load, or try series for a 16Ω load. In this way, you'll try the stock primary impedance, stock impedance/2 and stock impedance x2.
Maybe using the same type of speaker as your original one will help avoid perceived tone changes, which are really the result of a different-sounding 2nd speaker.
If you don't get earth-shattering tone changes with that big of an impedance change in either direction (-50%/+100%), I'd doubt a 25-33% change between 6k and 8k will do much to your sound.
Then the next place to look is the core size of the OT.
If you have 2 transformers, both rated for 6k:4Ω, one at 20w and the other at 100w, what's the difference? (Pretend you can find a 100w, 8k transformer)
There may be winding differences, but mostly the wire used for the windings will be a little thicker, and the iron in the core will be a LOT bigger in the 100w part.
Look at the claimed frequency response and what power level it is claimed to be rated at.
Many guitar amp OT's are claimed maybe 70Hz-10kHz, often at 1w or a part-watt. Hammond OT's intended for Hi-Fi use are generally rated for 30Hz-30kHz at full rated power for the part (100w for the 100w example transformer above). The wider the claimed frequency repsonse, at the higher power the claim is made, the bigger and heavier the core will be.
Go the opposite direction. What if you take a 20w transformer with 30Hz-30kHz claimed repsonse, and try to shove 100w through it? Assuming you don't melt the windings (and you probably won't), the frequency repsonse will be more constricted; you lose the extreme lows and highs.
Since guitar amps often have significant output tube distortion where we like to run them, and full response to the extreme ranges of those harmonics will probably sound harsh, the limited frequency response may turn into tone shaping. The transformer itself can also contribute its own layer of distortion, which may be subtle or dramatic depending on just how much bigger the input power is than the intended power handling. This is used as a tonal tool in clones of the Neve 1073 mic preamp (the analogy to guitar amps is somewhat limited because of the very different nature of the transformers used in a mic preamp versus those in guitar amps). Cut to the chase, 1073 users generally get a darker and more distorted sound when overdriving the transformers in their preamps.
So you may draw the conclusion that if you want pristine clean with wide frequency response, you want a big heavy OT that otherwise has the primary/secondary characteristics needed. A complex, distorted amp with limited frequency repsonse may call for a lightweight OT with a smaller core and otherwise the same primary/secondary impedances.
Example: My Standel 25L15 has extremely wide freqeuncy response and is clean as a whistle. Part of that is the circuit design of the amp, and another big part is the JBL speaker, which was originally intended for full range Hi-Fi use, not limited-range guitar amp use. But the 25w output stage also uses a 60w OT rated for 30Hz-30kHz at full 60w of output. I haven't investigated the parasitic characteristics of the OT to see if 25w duty actually extends the frequency repsonse even wider. Why such a big lump of iron, so overrated for the actual use? According to a trustworthy source who reverse-engineered an original 25L15, this OT was the closest match for the size of core used in the original. So this 25w amp uses a bigger, heavier OT than a Twin Reverb (8lbs for my OT vs 5.3lbs for the Twin).
What that means is the OT is removed from the equation in altering the amp's tone. OF course, the speaker I'm using could either be viewed as eliminating the speaker's contribution to the sound, or as a major contributor, because the speaker's frequency repsonse is 30Hz-17kHz. That's an enormous change from typical guitar amp speakers.
So bottom line, you may find after tinkering reflected primary impedance that OT core size is the biggest tonal variable when dealing with just an OT change. If you can ever find pictures (or an in-person example) of super-budget amps of a given power level and more expensive amps, note how much bigger the OT's are in the costlier amps. The small OT may contirbute to the cheaper amp's funky charm.
But amps are a balance of a LOT of competing factors, and the OT is just one of them (and often not a factor with the OTs many people choose).