Hi and welcome.
The .mod module format is really old and outdated. You really should not want to mess with it. Seen that there currently is alot of support for other module filetypes, like .xm and .it, I doubt that there isn't a library available that you can use instead.
There is another misconception possible. Although .mod stands for module, all other tracker formats are known as modules too. So an .IT file is also considered to be a module. If you're not the developer, you should verify if you can use .xm or .it files. (.xm is most likelly possible).
If it really isn't possible, then try to convince the developer to switch to another library that supports atleast .xm files. (or if possible .it)
The old .mod file really has limited support and not only will you get really crappy sample quality, you'll loose alot of abilities, like having instruments, panning per sample, amount of channels, length of a sample, mono samples, samples at a fixed pitch, no volume effects column, etc...
The only reason why those old .mod files sound decent, is because the crappy quality was taken into consideration when making the samples.
To point out what happened: Basically, if you're a painter and your colours are all messed up, but you know exactly how they're messed up, you can still make a very good painting with it. If you however use those colours to paint a painting the normal way, it will look really ugly.
So my advice is: go talk with the developer and see if you can deliver .xm or .it files (last one prefered)
There are alot of libraries available. I even seen .it support for a mobile phone game. If there's no support for other module types, you can also suggest .mp3 or .ogg support. That should be available too. You can always export your song to .wav and convert from there to any streamed format.