I've had 2 separate tickets at least a dozen times and have yet to have a problem checking all the way to the final destination.
I guess your mileage appears to vary more than mine in this case
If you don't have a printout of the itinerary though you have a 0.01% chance of getting it checked through. Same applies if you have an extended layover in the connecting city.
well I had as I said all of 1 time that I tried it on 2 different tkts
2 things going in the OPs favor however are FRA isnt a ST Hub and that hes flying back TLV-JFK on DL. I had a TLV-LHR-TLV & LHR-JFK-LHR tkts both on BA so they knew I was traveling thru but they also knew that by getting 2 tkts I was saving alot over a TLV-JFK-TLV tkt.
But from what I understood before I went with those 2 tkts or any set of tkts over just 1, is that the 1st Carrier does NOT have to tag the bags the whole way thru, if anything the opposite is the case. If however a person makes a stopover in route and the continuing flight is apart of 1 tkt then the 2nd Carrier has to abide by the Rules of a US Intl tkt, it doesnt have to follow the Rules of what the 1st Carrier would have premitted. The difference being anything over and beyond what the Agreement is for US Intl flights or 2 bags. So although I might be allowed 3 bags if I make stop then the 2nd Carrier has every right to charge me for my 3rd bag
So a NY-FRA-TLV-NY tkt allows 2 bags , while a NY-FRA is 1 bag on DL free in Coach 2 in Biz, while LH or LY etc FRA-TLV is 1 bag the weight depends on the cabin ones flying. However by having another tkt doesnt mean that the 1st Carrier must Tag the bag all the way all the 1st carrier must do is tag the bag/s to whatever is the final destination on the 1st tkt
I think the OP will have an easier job of convincing DL to tag the bags all the way thru, since they are returning with DL the whole way back and can say I only did it this way where its costing me more miles due to DL not being able to get me over there on DL or another ST carrier. Of cause DL can fire back then you should have cashed in for a Standard Award. But the OP is in a better state then if they had say a DL NY-FRA-NY and LY FRA-TLV-FRA set of tkts