Looking for advice on my specific predicament.

My flight to FRA (ex-DEL) was delayed by about four hours and will cause me to miss my onwards journey home (NYC). I've already been proactively rebooked to a new flight (SQ) but it's the following day but I'd prefer to get home earlier if possible. I found that FRA -> IAD is the only route with F availability on the same day. I called the LH F hotline and they said that the airport had issued the new SQ flight on LX ticket stock so they couldn't do anything about it (I have no idea why either). I called the first customer support number I could find for LX and after a lot of awkward silence and an escalation they told me that since I'm checked in they cannot change the flight. Since I'm stuck in FRA overnight I believe I'm entitled to hotel accommodations. What's the best way to chase this down in FRA? Since I'm arriving from DEL I expect I'll have to go through transit security to talk to anyone airside, should I just leave the airport and try landside support first? My inclination was to try FCL A or FCT thinking I'd get better support there but want to avoid security again if I can and expect I no longer have access to FCT since my flight is now on SQ and the next day.