What are some things an out of town sibling can do to support the in town sibling?

What are you doing that can be done long distance? Banking, paying bills? These can be done on line and might take some of the stress off you.
Having her take mom for a time would be a great thing but if mom has a problem with not being at home it might be easier for your sister to come for a visit and you get out. You can book a get away or even swap houses with your sister for a few days. Or if you are caring for mom but have your own home, own life just going home and not having to deal with things for 3 or 4 days is a big help.
Or simple as it sounds the simple act of making dinner every night when you are a caregiver is exhausting. Order meals, have groceries or a meal delivered. Just make sure they know that a delivery is coming. (hate to have a bag of groceries sitting by the front door all day.
Hiring a cleaning service 1 time a month or even as a 1 time deal would be appreciated. (maybe)
And hiring a caregiver 1 day a week or more would help you out a lot. You might have to ease into it with mom but she will get used to it. (have them start while you are still there all day and just do little things then you "have to run to the store for an hour" or go for a walk. It will get mom used to someone new.
This is something that she may have to get used to anyway as she declines and you need more help. You will not be able to leave her alone and there will have to be someone there 24/7

If sis will take mom for a week here and there, JUMP ON THAT! If mom can't really be alone, make up some kind of a little white lie about you NEEDING her to go since you are going somewhere or getting the wood floors refinished or whatever. Take the choice away from her since the break is for YOU not her. Ugh. Annoying.

Or can sis come to your place and you actually do leave? If mom won't agree to leave, then....

$ is always a nice help too. Unless mom can afford to hire a cleaning person, a helper of some sort, then sis could help with that.

Perhaps the two of you can split the cost on some care that helps during the week. Just a few hours a day (light cleaning, your mom’s laundry, change her sheets, and companionship, even) will take a huge load off you. I find the times I can go shopping and not have to worry, or just go to a quite place and not have to check on her and make sure she’s not hungry, or hasn’t fallen over….

And then, maybe you can plan for a week away to your sister’s still (you won’t have to pay for care that week), and when it will be hit and miss you won’t be nearly so burnt out and desperate for them to happen.