So, here is the thing.
Since you did not order the spare tire carrier with the camper, "technically" you can't mount anything there. The "warranty" runs out in a year so you could wait.
Having said that, if you use the OEM spare tire carrier, I don't see it being an issue; but it certainly could. Mount it close to the welded portion of the bumper and NOT in the center to limit twisting of the boxed bumper.
This is the one that came on my bumper from the factory:
Ultra Fab Fold Down Spare Wheel Tire Carrier Mount - $19.97
As to the bike rack. Go to plan B or C or D. The rear bumper on a light trailer is a recipe for disaster. Check on any of the hundreds of threads here.
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
HAM CALLSIGN - KC3FFW