I have the same problem, and took a long way to work on my bike to fit me. the options that would have worked for me are:

Integrated:
- Best: BMC (TM01 with flat bar and Profile Design Evo bracket for tilt, do not even consider the M/S sized one), Cannondale Super Slice (no tilt), Specialized Shiv TT new one (this bike only gets longer but not taller as size progresses), Specialized Shiv Tri new one (no tilt)
- OK but pushing the fit range: Cervelo P5 / P3X, Felt IA (but with the dagger stem only), Trek SC (technically in the not recommended range)

Traditional stem:
- Most flexibility
- Either add a -25 degree 120mm stem (a short -25 stem doesn't do much to stack but a long -25 degree stem also drops your stack a good 10-15mm)
- Use a standard 31.8 aerobar that supports a long pad forward position (Zipp Vuka Aero 2019, Profile Design Aeria Ultimate, Vision Trimax 2019, Look Aergo, Enve)
- Or use a standard aerobar with Profile Design Subsonic or with Zipp Vuka undermount

Best Bikes for bit with traditional stem to get long and low
- Best: BMC TM02 (Lowest and longest, but there have been multiple reports of fork snapping at the stem region on this very forum, so it wouldn't make me feel comfortable putting a long lever on it)
- OK: Quintana Roo PRFive, Orbea Ordu OMP, Cervelo P3, Felt IA

You can also look into Tririg Alpha, Vision Metron TFA and Profile Design Aeria Ultimate with its own stem. The TriRig was too short for me. The vision and profile design would have the pad bracket sticking out behind the pad in the extreme forward position, so I chose to get my reach with stem instead

I also preferred the traditional stem option because it also pushes the basebar forward, so I was also more comfortable when I have to get out of aero.

FYI, I am shorter than you, so you might have more or fewer options. I was looking for a solution for (Stack 570-580, Reach to back of pad 470+). I am not in a super aggressive position. Running a mid-stubby saddle and still behind BB. I just have hobbit legs for my height. I am running a bike with 120mm stem and pads all the way forward. Handling is not that bad, as long as you are not expecting road bike snappiness. For me at least, in a TT position, seat position have a larger impact on handling (how much weight I still have on the rear)

On a rant, my first bike fit had the fitter put my stack and reach at stack 640, reach to back of pad 370. A second retul fit put me at stack 600, reach to back of pad at 440. I was never comfortable wit either fit until I stretched myself further. Wouldn't have been possible with an integrated bike. In conclusion, I believe that non-integrated bikes are the way to go unless you have a fit dialed in already.

Adding a possible solution (not sure where your bike fit stands). I like to run a narrow arm position. My TT position (non-Tri) puts my arms close enough that a bottle won't fit in between. I found that I can tolerate a shorter reach if I space out the pads apart further. So for me, there is also a relationship between my arm width and reach preference. I can make a short reached bike work if I adopt a wider stance.