More power for moderate listening levels?

What is the volume level setting when you crank it up and notice the strain?  The XA-25 (same amp used in the INT-25) was tested by Stereophile at 80Wpc into 8 ohms and 130Wpc into 4 ohms. It does switch to class A/B at higher power output levels, so perhaps you are hearing that change?
I'm using the Pass INT-25 with Sonus Faber Electa Amator III speakers, 87db, 4 ohm, probably a more difficult speaker to drive than your Harbeths.  With my chair at about 8 feet from the speakers I usually set the volume in the range of 40 to 50 depending on the CD (63 is the max volume). I've also wondered if a higher power amp might drive these speakers better, although I don't think I'd want to give up the INT-25 for moderate listening levels.

There is nothing inherent in the design of an amp which will mean a higher powered amp will sound less good at low volumes than a low powered amp. The Pass Labs Int 250 may sound better or worse at low volume to your ears through your Harbeths than the 25. Whichever, it won’t be because of the amp wattage that there is a difference.  Whereas at high volume the amp output obviously is a defining factor. 

Great questions and comments.

Do the Harbeths sound strained at your normal listening position at high volume levels? 
This is a test I need to critically evaluate. 

What is the volume level setting when you crank it up and notice the strain?
Definitely can't get well into the 50's. Volume level in the 40's is still sweet and musical.

I think a big takeaway from this discussion is that sometimes you don't mess with a good thing.

At the SAME volume, will a higher powered amp improve the sound quality?

No. It does not work that way. As the great Robert Harley said, "If the first watt isn’t any good, why would you want 200 more of them?" Good question. You have a great amp. You have great speakers. Unfortunately way too inefficient to sound good together at anything more than medium volume.

What nobody seems to want to admit, at 86dB you need 2 watts just to get to 89 dB. Doesn’t sound like much. No problem, you got 25. But remember, that’s measured at 1m. Do you sit at 1m? Not even. Sound disperses at a somewhat different rate for different speakers but the inverse square is a good first approximation. That is to say, twice as far away, 2 squared is 4, you get 1/4 the volume. Let’s say you sit real close, just 2m away. Now you need 2 watts just to get back to 86, which is not very loud. To play 89 you need 4 watts. At 92 dB its 8w. 95, now just barely getting to what someone might consider loud, you need 16, and your amp is just about tapped out. 105 is plenty loud, but 10dB calls for ten times the power, or 160 watts.

These are the most conservative possible numbers. Sit even a little further back, in a normal room with normal absorption, and using average volume not intermittent peaks, you can easily wind up with ten times that, and there are not a lot of 1,600 watt amps. Which to me sounds about right for playing those speakers at satisfyingly loud volume. No wonder your beautiful Pass is strained.

A huge amount of our perception of "strain" is tied into volume. With seriously inefficient speakers such as these its just real hard to get the mind around the fact all that power is going into.... not much. Just wasted. Not even going into heat. The speakers can handle much more. Its just a bad design, for anything other than moderate volume listening.

Which, remember again, first thing I said, you have great speakers and a great amp. Just not great if you want to listen at high volume. Horses for courses.

I don't think you need more power, but you may need different power.

As a counter point, if you can listen to Ayre, Bryston and Luxman you should.