There must be some miscommunication of information here, because some of this isn't making sense. The "mismate" shot, given to stop a potential pregnancy in dogs, is basically estrogen and it must be given within a short time (72 hours max) for it to be effective. The prices for this - in the U.S. - are roughly $50-75 including an office call cost.
I'm sure that there are other ways to create an abortion/miscarriage later on, but the risks are high that you will hurt the bitch or even cause death. Why in the world would you want to wait? While the mismate shot does carry risk too (including increased likelihood of pyometria in future heat cycles, I believe), it is less risky than letting the puppies develop and then creating an abortive situation.
I'm also not sure on the age of the bitch. Somehow I have the impression that she's nine (I could be wrong on that though .. ). Letting a nine year old bitch have pups is dangerous especially if she isn't used to having pups. Now, there's a possibility she isn't pregnant, of course ... but there's no way of knowing until the pups are developed for enough for ultrasound or palpation. And, of course, checking for pregnancy will require a vet exam and the subsequent cost.
While I acknowledge that the OP says that she has no control over these dogs, she still can take information to her Mom if that's the situation. And the reality is that there are a LOT of reasons not to have these puppies and a LOT of reasons to spay/neuter, and very few good reasons to not neuter and to allow this arthritic bitch to have pups. Not only does neutering often help a dog live a longer, healthier and less stressed life but it prevents this problem of accidental breedings from happening. It takes very organized, very responsible handling to keep dogs from breeding when there's a female in heat. And when you have an older bitch and especially one that already has health problems (such as arthritis) you are resigning her to an uncomfortable and life-threatening time if she becomes pregnant. And with each heat cycle the likelihood (and probability) of pyometria increases. Pyo is VERY nasty and VERY dangerous - one of my chows ended up with pyometria at six years old and was extremely sick. I could have easily lost her. And why was she still intact? Well, I had shown her for awhile and considered breeding her at one point .. and when I decided against breeding her, I just put off having her spayed because she was easy to maintain. I won't do that again. It would have been 100% MY fault if she'd died. And, of course, the pyometria/spay surgery cost 5X what a normal spay would have cost.
Having Harry desexed right now won't change what he's going through at this point. He will still have the hormones that are causing him to be upset and excited - it will take a good month or so for those hormones to diminish enough to notice a difference. He will wake up, smell the bitch, and now be upset and excited AND in physical discomfort from the neutering. You would be far better off spending that money in getting the bitch taken care of so that she doesn't have to go through a possible pregnancy and the risks involved at this point. And then you can neuter the male in a couple of months when you will be "alright" with money.
By the way .. I have had many neutered dogs and I've never found it a problem to keep them in proper weight. All but one of my current girls are spayed and I've had neutered males in the past too. There have been NO problems from neutering any of my animals and a number of advantages (both physical and behavioral). The way I look at it is that when we humans accept the responsibility of taking an animal into our lives, then we are responsible for everything that happens to that animal for the entire life of the dog/cat/etc. And part of that responsibility is understanding and accepting that there are times we should choose to have things done to our animal in order to maintain physical and mental health and that often includes spay or castration. These are not aimed toward maiming or "cutting into" our animals .. they're a means of providing a better life for our pets. And that's a good thing.
Melanie and the gang in Alaska