None of the things you stated you have converts to Best Western points.

The only option you have, if you have no actual Best Western points, nor a Diners Club or AMEX Membership Rewards card (the only two point programs I'm aware of that can transfer to Best Western), is if this hotel shows up as a hotel bookable with UA, DL, or AA miles, which would be those respective airline websites. But be ready to find out that it costs way more miles than you think it should...

But at any rate, I just checked, and it's already sold out on July 2/3/4, it's only available at this instant for July 1 (for $226+tax, or 36000 BW points, per night).

I would not recommend in the future, when you see that a hotel is the "last" available hotel in a city for a parituclar weekend, that you "waste" time trying to figure out how to book it with points that you don't already have. If you didn't make a backup reservation with money, you presumably have to change your travel plans for that weekend (if indeed there's no other hotel at all in Seward left there).

Booking.com shows another hotel (Swan's Nest Inn) that's still available (4 rooms left supposedly) for July 2 (at $135+tax), but no one on July 3. (Keep mind different travel sites have access to different inventory at different indie hotels, so you should always check as many different hotel travel sites as possible if looking for as many hotels as possible in a hard-to-book location. And sometimes you just have to resort to calling hotels on the phone, since these third-party sites may not have access to every last room at a hotel.

But for that BW, it doesn't look promising for that weekend (beyond July 1), and the point requirement looks just as steep oin points (36000 is one of the highest point categories at BW) as it is on dollars.