When I try to record a file in the SeeMusic application, the damper pedal does not seem to working in the recording,

I see the same issue on my ipad

On MacOS, you can also trying changing the MIDI Input Framework from Default to "MIDIJack" in the Input Settings window.

Thank you for making the video showing the issue.

Have you tried recording the MIDI file in an app outside SeeMusic, like Garageband or Logic?

If the recording works fine in a 3rd party app, I would recommend doing the MIDI recording outside SeeMusic, then importing the MIDI file to SeeMusic after recording.