Flying with multiple airlines?

I'm flying with United and switching to frontier for the second flight.
My question is, if I check my bag I know I'll have to go out of security, to baggage claim, recheck my bag, and to through security again. So can I gate check my bag to avoid all of that?
Does it cost anything to gate check?

It's a carry on but I also have my purse, and supposedly I can only bring one of those on the actual frontier flight

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.

I would ask united this question

Last time I changed airlines, they transferred my checked bag from one airline to the other, and I did not have to go through security a second time.

You can gate check your bag but it will cost you, on frontier, you don't get a free allowance for any baggage other than a personal item, so if you take your carry on and want to gate check, you'll be charged $60, if you prebook a carry on $35 charge, they may still charge you the difference to gate check of $25. Frontier are low cost with loads of extra charges.

If you gate check your bag on a mainline aircraft, it ends up with all of the other bags in the cargo hold. You don't get it at the gate when you land you you do on regional jets.
Even if you are flying a regional jet, if you show up at the gate with something that is obviously too large to be a carry-on, it still could end up as a checked bag (with you being charged the normal checked bag fee).

Contact United and ask THEM if they will check your bag through to the destination even if it is on Frontier.
Many airlines have an interline agreement and will transfer bags to a flight on a different carrier.

That is the BEST way to know if you even need to worry about getting the bag at your intermediate stop.

Hard way