You should take your bag as a carry-on if you can, because you are correct that leaving to claim and recheck your bag will be a hassle.
As long as it is in fact a carry-on size, then you can take it as one of your allowed items on United (generally a carry-on and a personal item, unless you are flying Basic Economy, in which case you can only take a personal item). If you don't want to deal with the bag on the plane, or if the plane is very small, then gate-checking it is usually an option. However, if you are in Basic Economy, be aware that gate checking is not allowed - they will charge you the checked baggage fee PLUS an additional $25 fee.
However, Frontier will not "gate check" the bag for you. As a discount carrier, they charge for baggage. It won't matter if you check the bag or decide to carry it on, you will be paying for it. If you take it to the gate and try to check it there, then you will be charged $60. Better to pay for the carry-on bag now. They do not offer any free baggage other than your personal item.