Would I have any problem with the wheelbag as my only carry on item?
If there is a wheel in it, yes. I have tried this before and got shot down. They told me they would put a “fragile” sticker on them. My understanding is that you can only carry on stuff that will fit under the seat or in the overhead compartment. I’ve seen guitars as carry on and they let me take a carbon paddle.
I have done it on Southwest twice. From what I have heard you are taking a chance to get it on the plane for sure, but I hear more people who have done it. Could probably gate check it at the very least.
My strategy has always been to gate-check the wheel in a cardboard box (well-padded, of course). Ask the TSA agents at the security gate to hand-check the wheel. Once you’re at the gate, just seal up the box (don’t forget to take some packing tape), and let the gate agent know that you have an item to gate-check.
Gate-checked stuff is in a separate compartment from ordinary checked luggage, so you don’t have to worry too much about your $2K carbon race wheels getting crushed by those 50lb golf bags that are flying for free. 
Never had a problem with either TSA or the gate personnel.
Hope that helps!
cramer