Lumbar cramps, tension during effort

Interesting.

Generally lower back pain is caused by cramping of the muscles in that area.

These muscles stabilise / anchor the power being generated by the glutes and hamstrings... hell everything really.

As with any cramp, the reason it comes on is either overload, or a lack of required fitness/strength in the related muscle.

With lower back pain, this generally is caused because you are either isolating back muscles through having an overly acute hip angle, or because you are asking them to work very hard to stabilise your position... which can be caused by being either too stretched, having a saddle too high, or because your pedalling stroke is asking a lot of questions of your lumbar area (big gear, excessive push/pulling, etc.).

From what you explained... saddle far forward, I would not anticipate that your hip angle is too acute, which suggests that you are either too stretched generally, or your saddle is too high.

One additional thought is that with your cleats being set as far back as you can, you are shortening an important ‘lever’ which in turn will emphasise the effect of your crank length by reducing your ability to ‘ankle’. Ankling utilises more calf muscle, which in turn is less efficient, hence the move for many towards mid-foot cleat positioning. However, by pushing the cleats back, you will in turn be further loading, quads, hamstrings etc.

What length cranks are you using, and what is your saddle height / lay back.

I share your hyper-mobility and challenges, but just because you can hold a position, doesn’t mean you should. I am more upright than many I compete against, as I take the comfort / power generation over aerodynamics. 

If nothing works, try going the opposite direction - fully stretched out. Might surprise yourself 

Also, what bike are you riding and do you have a set back or inline seatpost?

The fact that you've got your saddle almost all the way forwards would send alarm behind ringing if this is on a road bike. Having the cleats and saddle as you've described would rotate your legs forwards, and create extra work to support your upper body. Especially as you are flexible I'd put the seat back in the rails. This effectively gives your legs more leverage to keep your body upright (pushing down at the front of the pedal strike pushes your body up)

For back and glute strength, exercises like deadlifts and squats are ideal, and I'd also look for IT band tension, either with a physio or by gentle foam rolling, as this can have impacts all over the body

At any rate, I'd go back to your bike fitter and ask for a bit more help, if their fit has led to other problems