Can I check "carry-on" bags airside @ Frankfurt switching to Lufthansa?

This is one of those questions that does not have a solid answer other than, It depends. It will depend on the gate personnel loading the plane. Since you are switching planes and not going thru a check-in counter, my guess is that you will be OK. My experience with Lufthansa at a check-in counter is that they can be pretty strict as to size and especially weight. At the gate, it is a different issue. And it may be dependent on the type of airplane for that last leg. If forced to gate check, it will be free since the gate personnel have no way to conveniently accept fees.

Are you able to access the Fare Rules and Conditions for your trip and find 'baggage' among the topics? There should be a 'black and white' answer there about which sizing rules apply and where.

Carry-on baggage is very 'class conscious' on Lufthansa. What fare class are you? One free checked bag is allowed for all but the cheapest econ class.

UAL has a 22" limit on carry-on bags. At 22.75" you are over the limit

You are on ice I would not want to skate on.

Sorry, I have no answer to the 'what does Lufthansa do at FRA' question

Hello
I’ve changed to Lufthansa in Frankfurt several times. The carry - on overhead bins are smaller than the NA carriers, I have found.

I know prior when I was flying with connections on different airlines with different weight/size limits, I was told it is the 1st airline weight/size limits I would need to follow. However, I haven't flown with Lufthansa. You can probably contact Lufthansa and ask them, providing the information that your 1st leg is UA. So either Lufthansa honors UA luggage rules, or you need to have luggage that meets both their requirements. I'm not seeing you leaving airside to go back out to check-in at the counter. That wouldn't make sense.

If Lufthansa honors UA rules and your luggage doesn't fit in the overhead bins, I'm sure they can always gate check your luggage.

With that many connections, I'd check from the very beginning...especially since you will have to on one leg. For me, I'd rather not schlep a lot of baggage through airports. I've been on so many flights where people are forced to check bags that I'd like to avoid "bin" anxiety. On planes, I prefer to have just what I need. Pick your poison:

The only time in 40+ years of international travel that my luggage has been weighed was for a Lufthansa flight from Chicago to Frankfort. My suitcase was a small "international" carry-on--19" x 13' x 8"-- and weighed 18.5 lbs. according to their scale. I would not take the chance if keeping my carry-on with me was important.

We use RS carry-on, and have gone through FRA both ways. Never weighed here or there (check at home, we are always under 18 lbs), and this ebags will clearly fit the sizer. Only time this was ever an issue was when the gate agent in FRA stopped my wife, just under 5'2", and told her by eyeball the bag was too big. I had just passed before her with the identical bag, no question. Of course, it looks smaller on me since I am 5'11". I immediately confronted her with something like "shall we switch bags and then you can tell her the one that was okay for me is now too big?"After a couple of brief annoyed bak and forth, we simply continued on our way (her pass and already been scanned), and they did not pursue this. And of course the bags fit fine on the A319. Now in our case, the bags are on our back, we are walking comfortably, and no one should ever question this. Rolling your bags, I am not sure. We have all seen a lot more than a spinner that may be barely too big and a backpack on top of it not stopped.

I would not count on Lufthansa "honouring" United's requirements.
In 2019, I flew Air Canada to Italy with a carryon of 20 ", they did not weigh it.
On return with Lufthansa (same ticket booking) I was very sternly reprimanded in Florence by Lufthansa for having the "wrong" size of carry on.
They were going to make me pay to check it, but it was a busy day, so they checked it for free in the end.
Be prepared to pay!