The problem here is that this sort of RV air conditioning unit isn't designed to be recharged. There are aftermarket kits that you can fit to them, and the video Lisa found made it look fairly easy to do. There's a relatively inexpensive part that you can use to tap in to the piping. Lisa said the video could have been using her AC unit, it was so similar. We hunted around online, and then I thought of checking the appliance-parts place in Reno where we got parts for the washer before we gave up and bought a new one. Lo and behold, their web site says that they have 50 of the things in stock, so we'll go in tomorrow after work and buy one.
Coolant is a different story. Lisa's travel trailer is old enough that it uses the older R-22 coolant, which is banned from manufacture after 2020 and rather expensive. However, you can still buy it, so we've bought a coolant replacement kit (includes hoses and pressure gauge). If it works, we might buy a large bottle of the R-22 coolant to serve as a lifetime replacement supply.
Why not just replace the whole AC unit, you may ask? Well, Lisa was not that thrilled with the look of replacement AC units when she checked them out earlier this year, and would rather repair than replace. Even considering the higher coolant cost, it's still cheaper to repair than to replace.