You might have to do some trial and error which may include the uncoated profile. Your screen mesh may have something to do with the amount of ink getting onto the shirt, too.
Since screen printing lays down a lot of ink meaning there is a lot of gain on press you'll have to do some experimenting using an assign profile procedure that might get you a CMYK profile that gets you in the ballpark. Or at least it will allow you in coming up with your own custom profile within Photoshop's Color Setting CMYK menu that delivers numbers that clearly show that dot gain is being compensated for on your press.
You'll use the dark print and the numbers within the file to act as a reference. Scour the web and download as many CMYK profiles you can find that are associated with presses with high gain like newspapers and even screen printing. Who knows maybe by now someone has already gone through the same thing and came up with one that works. The numbers when converting to the newly found high gain profile will be smaller compared to the Web Offset reference file for instance 50% dot will read something like 30-20%, 80% will read 60% and so on.
Once you've collected enough CMYK profiles start assigning them temporarily for preview purpose to the original CMYK that produced the dark print. The one that makes the image the darkest is the one that will get you in the ballpark. Notice the degree by which it goes dark and pick a profile that produces the same degree of darkness that showed on your press.
Noting the degree of change during this assign procedure will also aid in coming up with a custom CMYK profile in Photoshop you can later assign to the reference file in making it look just like the dark reference print.
It's not very precise but the only alternative is to have a custom profile made using a spectrophotometer which can get expensive. You might look into ColorMunki which uses a spectro and is around $500. Not sure if it can produce CMYK profiles though.
Thanks for your help. It sure points me at the right direction. In the short term (to solve this problem) I downloaded Fast Films trial version and that really did a nice job for me. It also gives me another reference point to work towards with your answer below!
Bell Binders Co.