Binning should not modify the FOV that is pre-calculated and provided as a hint to the solver.
In the logs you provided I see that a second solve is started before the first finished. This prevents source extraction because the input FITS has a hardcoded name, not a temporary name, and the solvers fails systematically. This could be reported as a bug on the INDI github, but it's unexpected that a second solve request be requested before the first finished. However, I observe that you don't wait for the second solver execution to fail before disconnecting indiserver.
For instance, in
indilib.org/media/kunena/attachments/4725/log_07-30-39.txt (this is CCDSimulator on a 24.4 arcmin FOV), a first CCD capture is forwarded at 7:32:20, solver starts, and then a second is forwarded at 7:33:26, before the first solver procedure is finished. The first solver procedure breaks because apparently the solver accesses the FITS file sources once per index file. The second solver procedure is aborted by the disconnection of indiserver. So in that log, I don't see any solver failure, only interrupted solver procedures.
On the opposite, in
indilib.org/media/kunena/attachments/4725/log_07-27-28.txt (same CCDSimulator on a 24.4 arcmin FOV), the first CCD capture is forwarded at 7:29:32 and solved successfully at 7:30:18 with a pixel scale of 1.78"/pix.
I have no straightforward explanation for the duplicate capture occurrence. What I see (and you see that too of course) is that when you run a CCD simulator capture binned at 1x1 with GSC, the solver succeeds. So I'd stick to that configuration to successfully solve a frame in that configuration. 4x4 binning on the CCD simulator using artificial stars is not giving satisfactory results, that's all.
In your last run with a configuration matching your Atik, GSC produces a ~64.4 arcmin radius frame. Same observation as you, 1x1 binning works, 4x4 binning doesn't have time to complete. Here again, no failure, but an aborted solver procedure. However, the CCD Simulator rejects your width of 3362. So I guess from there the frame doesn't match the expected FOV anyway.
Note that while I don't observe failures, I'll admit that a solver procedure - in your situation - taking more than 3 seconds is actually bound to fail.
Even with these failures, I believe that your setup is ready for a live test with your real equipment. Do it simple, sync 15° from pole, bin 1x1 and let the solver find the solution.
-Eric