I am using Joomla 1.5.15 and am upgrading Multisites to 1.2.62. I have 16 slave sites running. After much trial and error I am able to get things running smoothly but after a short amount of time everything goes blank. White screens appear when trying to pull up any site or any admin area.

If I remove or rename includes/multisites.php, then the sites all run smoothly. At this point I am able to only login to the admin for the master site, not any of the slave sites. But all sites run fine on the front end.

Any idea where the incompatibility could be or what I could do to fix this?

As mentioned plenty of times in this forum, a blank screen is the symptom of a PHP fatal error. So you should check your "error log" file to get more info on the reason of the error.
Perhaps you have a missing file somewhere or a permission issue or other reason that I don't imagine for the moment.
So try to get the info on the reason in the "error log" file.

I have checked the error logs several times and cannot find anything in there.

When I retry the install I do get two generic Store errors. But the plugin says it installed so maybe that's the issue that a file is missing or not overwriting properly.

Can you give the exact error message.
I searched for the "Store errors" message everywhere in Joomla and I didn't find it.

If you can add a screnshot, this will be helpfull.

I don't think that this error is related to the files and folders on the disk.
I suspect that this error message is related to a record in the DB that can not be written.