Can you review the server error logs to find the causes of the errors?

You can use a network sniffer (e.g. wireshark) to capture the traffic between your system and the server. You might find some clues by comparing successful and failed accesses. Even with HTTPS, not everything is encrypted.

There are tools that might allow you to inspect the HTTPS exchange with the server. If Firefox browser works, you might be able to use some of its debugging features to find out what is happening. Otherwise, there are proxy servers which might allow you to inspect the exchange (e.g. fiddler. Ultimately, you could brute force decrypt the traffic, thought this might take some time.

Might cookies be required by the server?