I towed a 31.5 ft TT with my 6.5 ft bed super cab 1997 F150 with a 5.4 and 3.55.
The motor lacked the power of today’s trucks but the chassis was relatively stable.
I eventually upgraded to a long bed Dually and my tow went from relatively stable to a complete non issue. Even with a heavy golf cart in the bed the Dually barely squatted while towing.
I would not want to tow a 37 ft TT with any ‘Half Ton’ with the exception of the Titan XD.
Will you be relatively stable? Sure...you can make it work... but your looking at the wrong truck for a 37ft trailer. JMHO based on experience.
Thanks!
Jeremiah