OK, you have nixed virtually everything I have suggested. I'll give it one more try, and then shut up.
Yes, private ice is expensive. But if all you want to do is to have a friend video a 5 minute routine to send to a remote coach (which might perhaps work, for example, through ICoachSkating.com, MySkateCoach.com and OnlineSkateSchool.com), work through the numbers: Assume a rink is willing to tack on a 5 minute extension to an existing session, and that they charge $300/hour. Then that 5 minute extension costs $300*5/60=$25 - probably less than a private lesson, or a couple club or freestyle sessions.
If you can do the routine during a club or freestyle session, and pay a coach to video you, thereby possibly meeting the rules of your club, that might be cheaper - but maybe not if you include the cost of the session, and the cost of the coach. Also, assuming that your club or rink enforces strong enough right-of-way rules for you to capture the whole routine on video during a club session.
Either way, you also have to pay a remote coach to evaluate that video. The number of coaches who are good and experienced at working remotely is limited, and the number of good pairs coaches is also limited, so they might charge more than your freestyle coach, probably more expensive than the ice time. Oh well.
There are lifts and throws in ballet, modern, and contemporary dance, and at the higher levels in ballroom, swing and hip hop. Most teachers and dancers in those areas will not know much about the rules and standards of pairs skating, but perhaps they all worry about safety. Perhaps they can help?
If you can deal with safety that way, perhaps you can hire a choreographer instead of a coach. A lot of them work remotely. Including some of the coaches who work through the sites I just mentioned. Many skaters send videos of their skate routines back to the choreographer for comments, as you might want to do, which is sort of like a remote coach.
E.g., Audrey Weisiger teaches
Grassroots to Champions workshops to budding young skate choreographers, and might be able to choreograph for you, or suggest and put you in contact with one of her students for the purpose. She is a very pleasant person to talk to (I met her briefly). Nick Perna advertises that he teaches choreography too. Both work through those sites, and Audrey also advertises through the Grassroots to Champions site.
(They are also both
expensive elite coaches - Perna even teaches pairs - so they don't quite fit what I was thinking of. But if either puts you in touch with one of their choreography students, the student might be cheaper.)
Of course, none of this is going to be free!
BTW, even though you already have a partner, you might still be able to use a partner search site, to locate other pairs skaters and the coaches who teach them. The skaters can tell you who coaches them, and discuss other issues with you, and the coaches may be looking for other students, possibly including remote students. In any event, good coaches aren't all that hard to find - e.g., you can easily look up who coaches the skaters who win semi-local competitions, or pass semi-local tests.
But you have chosen a sport which is always going to be somewhat difficult in your location, for your schedule. And if you later decide to coach it yourself, there might be a very limited clientele in small town America. As an athlete, you must love challenges, but this isn't a challenge you overcome purely through training and practice. But of course, you knew that going in.
If you test or compete your routine, you will still have to travel to a big city to do it - unless a local judge knows how to judge pairs. (If one does, most judges, AFAIK, start out skating, then coaching, and might be able and willing to help - though, last I checked, USFSA judges have conflict of interest rules, restricting their ability to act as a paid coaches. ISI rules may be looser - I'm not sure.)