Usually the bypass only needs to be active for starting, its not part of the bypass module that keeps it running. Generally the cut off if a safety feature of the remote start module to prevent someone from driving off by smashing a window.
Workarounds are put in a separate ‘push button start’ kit from amazon/eBay - they run about $30 and don’t have this ‘feature’, or alternatively, just put a physical switch for you car’s ‘on’ key position on your dash.
If you did this, it’d be remote start (triggers the bypass module), flick switch (holds ignition on), put in gear (car stays running, enabled by switch), when you’re done driving, turn off switch (car turns off since both remote start and switch are off)
And that way you get the safety of not being able to start with the switch (doesn’t trigger the bypass, doesn’t connect to start wires)