Booked a handful of trips CMH to TLV (and back) last few months and had some similar issues. Overall thoughts, some which may be not helpful, but are rattling in my brain here:
1) 27 hours AMM to BNA seems brutal on this itin and I think your itin flies through ADD, which sounds fun. But the 93 miles is a good value for business class, so maybe so!
2) I was playing with your itin and saw an itin that might be more attractive. Book in/out of IAD and you get flight there direct or through FRA on LH and direct flight home on United from AMM (leaving 1 am on 5/8 and direct to IAD) in business on the way home which was 3200. I saw a 2800 fare booking on united direct to TLV (econ to TLV, business AMM to IAD) and play upgrade roulette. I think your upgrade chances on LH are much higher than United but would be a lot of PlusPoints since it is two legs. Last few flights I have been on LH, upgrades seem to clear, though very close to departure. On UA, upgrades to TLV have been hit or miss for me. Two of us and one gets an upgrade, sometimes, and I am GS. I have not seen prices on flights to TLV go down closer to departure over the past year, but ymmv.

One other option, buy IAD --> AMM roundtrip direct flights if you depart IAD the day before. Economy on way there, business on way back is $3000. Bet the upgrades IAD to AMM are more likely than IAD to TLV

I think that Z class AMM to IAD is a steal of a price on that day if you book it round trip and willing to pay.