Sounds like you at least have a direction to work in. If you raise the rear more and like the feel of the bike in the corner but feel like the rear lifts under braking, try what Rob said. Sounds like the bike is pitching over center too much or too fast when you go to the brakes. Things you could try: Add preload to the front Change valving in front Raise fork oil level in front Go to the rear brake just before you squeeze the front brake lever. This can help settle the rear. Raise the front. You may need extended fork caps. I know it sounds like raising the front would put you right back where you were, but not necessarily. Swingarm angle will be different with bike higher and it should have more mechanical grip.
Keep going in the same direction if that helped it. Just do it from the front, either take preload off the front or raise the forks in the clamps lowering the front. The front is too high at that moment and is causing what you are feeling. When it gets too low it will feel like it wants to fold in mid corner. A track with elevation changes will highlight what area you need to work on if you pay close attention. If it turns in better in downhill stuff where the front is loaded more than uphill stuff the front is too high and vice versa. Its all about balance, finding a combo that works for you everywhere. Of course then you start going faster and it all changes again....
I don't know about the 05's but the previous gen liked the rear raised and the forks dropped. My 02 750 is shimmed up 10mm in the rear and forks are pulled up 10 above the triple.
That's what we did w/ my 1K....kept adjusting/changing heights, then re-adjusting heights, etc, for years. Chasing ghosts. Get that thing on a geometry machine, then it's Done.
IIRC working with Mike back in the day, the stock top clevis couldn't give enough ride height(shimmed to no threads left)
Are you blowing through front fork travel on brakes? This can cause the rear to lift. Might need to recheck the air gap and add fluid 5cc at a time. 5mm at the rear shouldn’t make the rear lift that much.
I had a good/local suspensh guy refresh my forks in 2016, and he told me he put in less oil. I was just riding on the street, but was instantly on the stops and carrying the rear wheel, w/out trying/wanting to, every time I was on the f brake hard, regardless of speed. Oil level is what I would look at. I sold the bike a couple months later and told the guy to add oil to the forks.
Raising the rear more directly addresses the running wide on throttle and the swingarm angle. It indirectly addresses the turn in issue.
There are two I have worked with personally and would definitely recommend. https://www.thermosman.com http://gmdatl.com/index.html GMD Computrack has other locations around the country too. There are others out there too and I'm sure the beeb will chime in with their info and experiences.
a great suspension tuner should have base-setup numbers for your bike. spring rates, fork ride-height, shock length, chain length, etc. they have probably worked directly with a racer on that bike to optimize it all. that experience is a big part of what makes them "great". if a tuner gives u a funny look when u ask for what shock length works best for your spring and your tires, Id take my $$ elsewhere.
If your close to St Louis I'd recommend Velocity Calibrations http://www.velocitycalibrations.com/ . They have a Scheibner CMS that can measure up the frame and tell you if it's out of line, then help you set up the bike.
06 I think started with longer caps. On 05 forks can be extended internally. I had 05 750 and if I remember I had spacer under clevis and +5 on shock. Forks also around +5. One of my favor bikes, good handling.
When I bought the 02 750 it was set up as it is now. I stupidly put it all back to stock as I went thru the bike then after the first weekend at the track I ended up going back to the set up it had when I bought it.