Here's the update.  Prior to taking the plane to the shop to have some
avionics work done, including overhauling the AI, I decided to try a
barrel roll.  I found another suitable location on the sectional chart
while waiting for the oil to warm up after engine start and made my
way out there. After clearing the area, I attempted one to the left.

Looking back, I realize that drawback is that I didn't have a clear
idea in my head of EXACTLY what I wanted this to look like in terms of
the initial pitch and the roll rate throughout.  I should've done a
much more familiar maneuver, something that leaned more towards an
aileron roll (although not a perfectly pure one) than a big barrel
roll. I don't think I got the nose up high enough, and I was rolling
too slowly.  Consequently, approaching the 135deg mark, my nose was
lower than I wanted.

I had also not reconciled the difference between the control pressure
required on the aileron vs the elevator. With 20/20 hindsight, the
quickest solution would've been a quick roll to the right to recover.
Instead, I made a decision to work with the elevator, the most
comfortable and natural movement on the stick at the time (I know it
sounds odd, but if I'm being honest, that's what happened). I pulled
throttle all the way back and completed a pseudo split-s instead.  My
primary concern was not overloading the airframe and not approaching
Vne.  Aircraft weight was around 1630, which was in my favor. I
completed the split-s, deliberately limiting the G's to just a shade
more than I'd experienced at the bottom of some fairly tight
descending turns in the past, while keeping an eye on the speed. ASI
showed 215-220 as I finished up and transitioned to a climb to bleed
it off.  I could've pulled harder, for sure...but I figured that if I
kept the G to a familiar level, while not exceeding a comfortable
speed, it would keep things as familiar as possible.

Furious at myself for not having a proper exit plan, and half-assing
the roll, I got back up to altitude, re-cleared the area and did it
again, with a much more reasonable roll rate.  It was faster than a
Bob Hoover barrel roll, but still kept positive G the whole time. It
went according to plan.  It looked closer to an aileron roll than a
barrel roll.

I decided to knock it off, and go back to debrief on what happened and
why during the first one.  I knew the ailerons were stiffer than the
elevator, but I'm surprised that it affected my gut decision when
things weren't going to plan.  Next time I will command the control
input required, regardless of stick feel.

It was definitely a learning experience, and I'm glad I did it,
although I don't think I've ever been more disappointed in my flying
and planning.  I'm happy with my thinking and flying during the
split-s, balancing attention between G and accumulating speed, but
absolutely floored that I got into that position to begin with.

This was going to be a 'one off', based on my impression that it was
not great for the AI. However, when I dropped the plane off at the
shop, I discussed it with them. They were of the opinion that smooth
rolls 360 deg rolls were not a cause of concern.  Sudden, abrupt
movements, 'slamming the gyro', and spins were the things to be
avoided.  So, now I've heard opinions that cover the whole spectrum.
I think I'll call an instrument shop (that's where the AI is going to
be sent, the current shop is doing other work).

So, in short, the learnings:
1) know precisely what it is you're going to attempt to do (jeez, I
feel like a tool even saying that one), and have an exit plan
2) have WAY more altitude available than you think you need. A roll
doesn't need much (actually, any, if done properly), but a split-s
sure does.
3) ignore what's 'comfortable' with the stick. A lot more pressure is
required to work the ailerons than the elevators, which is a departure
from other control systems with which I have the vast majority of my
experience
4) this last one isn't a learning, just a confirmation of what I knew
and already heard from many ppl, the speed builds FAST when the nose
is low. There isn't much time to bugger around when the plane is
heading down.

Whether I pursue this any further will come down to whether this
really is bad for the vacuum powered AI, or not. Right now I have a
slew of opinions, so I figure my best bet is to speak to people who
actually maintain those instruments.

Keith