Well, I have done something similar several times (ZRH-JFK with arrival at T1 then travel to T4 for a connecting flight out with ~100 minute connection time). In this case my connection was on Delta which is of course a separate booking so completely unprotected.

Overall, I think it depends on a few things - are you arriving on LX16 or LX14? Immigration and customs are significantly less busy for the earlier flight. In F you will be first off and should not have much of a wait if you're on the early arrival. If you happen to have Global Entry that minimizes this concern but I've had colleagues in F who did not have it get through the immigration lines faster. If you're on the later flight it becomes more challenging but still possible.

Are you checking luggage or hand baggage only? Delays on arriving bags are a real thing here and the F/HON delivery is spotty. Best case it will be shortly after the crew bags.

I do not recall if there is a COPA desk for re-checking bags in T1 or if you'll have to take them to T4 to check there.

AirTrain to T4 isn't bad - maybe 5-7 minutes? Security at T4 is more efficient than T1 but depends on whether you have TSA PreCheck or not.

Overall, you're PROBABLY ok but it depends on a lot of unknowns that can shift it one way or another. Depends on how much risk tolerance you have here.

As for how much assistance LX will provide? My experience is not much at all. The only help I've seen is someone who stood at the baggage claim and watched while I got my bags off the belt.

The question about how helpful they will be if you misconnect due to customs/immigration/security is a good one. In other situations (not this, as I was on my own), they have been helpful to me but not terribly helpful to others. Problems due to "administrative issues" have been dismissed as "not our problem - sorry" as much as I've seen them be helpful.

I hope this has been a little bit useful for you - I'm happy to provide more detail if you'd like.