Integrate geometric txmgr with avs-sync

[samlaf]

Is your feature request related to a problem? Please describe.

We just merged the geometric txmgr in #284, which has unit and integration tests. But we should still test it out with a real service using a real network.

Describe the solution you'd like

Integrating it in https://github.com/Layr-Labs/avs-sync would probably be the easiest. eigenDA is currently running avs-sync, so we could test it there.

Describe alternatives you've considered

If any avs is willing to test this it would also obviously be very useful feedback. :)

Additional context