Improve Duration of Integration Tests

### What should be done?

The integration tests should have their runtimes reduced without losing much / any coverage.

Tests include:
- Integration tests ~12min [example job](https://github.com/0xMiden/miden-client/actions/runs/15242716980/job/42869427275)
- Integration tests for web client ~27min [example job](https://github.com/0xMiden/miden-client/actions/runs/15242716980/job/42869427448)
- Integration tests for remote prover ~16min [example job](https://github.com/0xMiden/miden-client/actions/runs/15214169337/job/42795504580)

### How should it be done?

Unsure. Needs investigation.

Perhaps
- Improved caching
- Test fixture reimplementation / restructure
- Parallelization (separate jobs)

### When is this task done?

When we agree on a maximum duration integration tests should take.

When the integration tests take approximately that amount of time.

### Additional context

_No response_