Best amplifier for klipsch heresy 3 speakers.

If you want a transistor amp that sounds any good on that speaker, its going to make some heat too. If you don't mind compromising, then any amplifier will work.

The 'tame on highs' thing is where you are most likely to get in trouble with most transistor amps. Transistors have non-linear capacitive elements in their junctions. This contributes to distortion. It is also magnified by greater amounts of current through the device. To control the resulting distortion, feedback is often used. Both contribute to odd ordered harmonic distortion, which the ear interprets as both brightness and harshness.

That is the physics and the physiology of the situation. If you are dead set on a transistor amp, put a level control in series with the tweeter so you can tone it down. It won't be ideal but it will work.

Alternatively, you could get a speaker that is designed to work with transistors. The speaker you have now does not sound right with transistors because it was designed on and for tubes.

Anything from Cayin audio.

I've enjoyed Musical Fidelity integrated amps with my Klipsch Forte's. Deep tight bass, nice mids and detailed, but not bright highs.

If you are really uncomfortable with the sound of the Heresys then perhaps the better part of valor would be to sell them and start over. I realize that you may not recover all your funds but the money you'll spend trying to get them to "sound right" may add up fast.
I never gave up myself with my JM Labs Electras and eventually found the right combination of gear to make them sound great.
I started by telling you that you may want to get some used Cardas cables, that had a big impact on tone. I eventually found that Tube power amps did the trick in combination with a specific type of tube pre amp. But it took 5-6 years to figure it out and to budget enough money to do it.
There are a lot of inexpensive tube integrateds out there that you should keep an eye on. As mentioned Cayin. Jolida, and I will add Prima Luna amongst others, but I thought you said no tubes? An older British tube amp from Quad might be the right thing. You will only know if you actually try it in your system, there is no really quick cure in most cases of "Trebelosis extremis".
Do some homework over on the Klipsch Forums as suggested for more ideas and advice. BTW I have my LaScalas attached to my old small SS Mac even though I have several tube amps I could easily use instead!

I don't understand the issue with a wife working out in the same room as a tube amp...but, of course, I can guess: Sweat beads landing on tubes causing them to crack, a large plastic exercise ball melting and popping from hot tube contact, tube glow reminding her of fading love and vanishing youth, she prefers there be no euphonic harmonics taking the edge off of Joe Scarborough's innanities, there are cables running under the Bowflex, she thinks "I have to get in better shape so he spends more time with me and not this stupid hifi hobby", splashed chlorine from the lap pool is making the amp smell like a small town YMCA locker room, extra heat from the tubes are causing the emergency food rations stored in the basement to rot...or something...