Thanks for offering to delve into this problem in more depth.
The best way to debug the KeePassRPC client (Kee extension) would be to use the debug version of the extension. If you’re using a beta version of Kee then unpacking the crx would give you easy access to the source code for that purpose, or you can download the zip file from the relevant GitHub release page (this would be the zip file that Google then sign as part of the conversion into the crx file). Typically, I would use an unpacked version of the debug (beta) extension which I have built from source on my development machine but if you use an unpacked version that the CD pipeline in GitHub has built and attached to a release, you’ll get the same in-browser experience. In fact, even just using the beta version without unpacking it should work too, although that’s not something I have done very often.
Once you have that working, you can use Chrome’s inspector to debug the extension’s background process and thus see the source code (via the source maps that are included only in the beta version of the extension) as well as interact with the running state via debug breakpoints and logpoints.
The SRP one-time password dialog is triggered in the KeePassRPC plugin when it receives a connection request from Kee and can find no associated shared secret for the unique ID that Kee supplies as the “username” for the SRP connection (broadly speaking). Thus, I expect if detailed debugging of the SRP protocol is required, you would be best off focussing on the C# code in the KeePassRPC GitHub repo.
KeePassRPC technical detail could also contain some useful information without you needing to dive deep into the C# and TypeScript implementation code.