Minimum Connection Time at DOH on a through ticket is 45 mins. Checked baggage or not does not affect that.

If you want to risk it on separate tickets, check in for the second flight online, and hope you don't hit one or more of an inbound delay, a distant bus gate arrival, busy security, and another bus gate or distant contact gate for departure. You need to be at the departure gate min 20 mins before departure. If they know your inbound is >5 mins delayed you may be offloaded anyway.

I do a lot of separate ticket journeys but I wouldn't risk that schedule on separate tickets, sounds potentially rather expensive YMMV of course.

To add, my flight last week was scheduled to land at midnight. We arrived in the area on time but ATC put us in a hold pattern and the arrival time on the moving map jumped back by 25 minutes. Not the first time that's happened, midnight is very busy time at DOH.