Relax!
likelihood that TG will check through on separate tickets of a non alliance partner: 50% (it would be 100% for a separate ticket of a Staralliance airline)
in case they will not check through
domestic arrival and pick up luggage: normally 15-20 minutes, max 30 minutes
take elevator to international check in area and walk to check in: max 10 minutes
so you have then 3 Hours 45 minutes for normal international check in, outbound immigration and security, more than enough