Will short arm crank conversion work for me?

We cannot tell you what size cranks you need.

Perhaps someone has direct or similar experience to help you with. If not, here's my advice ...

In order of preference

1. Talk to the rehabilitation people who (I am guessing) helped with the knee replacements. Find out if they can advise you on this, or if there is a sports physiotherapist (or similar) that they recommend. Having those people talking to each other usually fast tracks the process. At the very least I think you need to know the likely consequences of your choices. Is OK to work through some pain or discomfort? How would you know that something is wrong? The more info you have the better.

2. Find a sports physiotherapist for yourself. If you already know one and they can't help, get a recommendation to someone who can.

3. As a last resort, get a friend to support the bike while you test what movement you have. Make marks on the wall or use some other means of measuring what you can do without pain. Check if both knees have the same movement. If they are the same then half the distance that you measured is the crank length you can use. Make sure you understand the idea here before going further. Then go back to the shop and see what cranks they have that you could use.
   The idea is to have an independent idea of what you need before you talk to the LBS guy again. No matter how honestly helpful he is, be prepared to get a second opinion as well.

Good luck!

Update

Web searches show that cranks can be found in many non-standard sizes, such as 130mm. Some people will take longer cranks and cut them down to the size you need. I'm sure you'll be able to get what you need! The first step is to work out just what that is.

Where your LBS could be of great help is working out what gearing you'll need with your short cranks. As you get stronger your needs may change, so keep up a good relationship with them.

My dad had this exact problem (Just one knee). My first attempt was 150mm cranks. No where near enough, but worth the $30.00 (Cheap kids bike part) as an experiment.

Next I went to crank shortener. He started at equivalent of about 130mm, and is now up to 145. Since then he switched to using an electric bike - his crank shorteners moved with him to the new bike (the different Q Factor is not a problem for his casual use). We though about drilling/tapping a longer crank to a shorter one, but decided for his use the benefits are not worth the cost (And are quietly hoping to get him up to the 150's). The electric bike has funny right hand crank that does not help.

More expensive options include pivot cranks and "Power Cranks" and are worth investigating is money is no option.

I would invest $120 or so for crank shorteners. This will establish what length you need and allow adjustment as your knees improve. Stay on the shorteners till you feel the improvement has stopped, then consider purchasing/fabricating cranks of the correct length. If you invest in short cranks now, you may be buying longer ones later.