How do I listen to music on my computer through my iPad?
Here are a few possible solutions:

Plex

I haven't used this but it appears to do exactly what you ask for. Install the free server on the Mac and the (paid) app on the iPad. You don't need to sign up for a subscription and it says you don't even need to sign in to stream from a local server, which should help assure you that no streaming from the internet will take place.

Rogue Amoeba's Airfoil

Install Airfoil for Mac on the Mac Mini where your music is and Airfoil Satellite for iOS on the iPad. 

Airfoil is not free, but you can try it before you buy and it should work reliably and is well supported. It routes any audio source(s) you choose on the Mac to play simultaneously on one or more compatible destinations - so you can use any Mac audio player you like, including but not limited to iTunes. Airfoil Satellite has basic remote control capabilities for some applications such as iTunes and Spotify, but you would need something else to create and edit playlists - I'm pretty sure Apple's iTunes Remote can do this, for example, or if you use VLC on the Mac there's VLC Mobile Remote for the iPad.

Logitech Media Server

The server you run on the Mac is free, on the iPad you would need the SqueezePad client app (paid). 

DLNA

Install a UPnP/DLNA media server such as Kodi (free) on the Mac and use a DLNA client app for iOS - I don't know which to recommend, but there are lots to choose from including some free ones.

iTunes

You say you don't want to use iTunes and iOS Music because of the risk of it using too much data. I'm not sure whether you're talking about iTunes on the Mac Mini downloading purchased or matched (if you have iTunes Match) tracks, or Music on the iPad streaming these tracks or playing from Apple Music (if you have Apple Music), but there are two ways you should be able to use Music on the iPad without this happening: 


Sync the music from iTunes on the Mac to the iPad via USB. You will need enough storage on the iPad to fit it! In Music on the iPad, make sure you only play from within the Downloaded Music section of your library.
Turn on Home Sharing on iTunes and only play from within the Home Sharing section of your library. I (and others) have never found Home Sharing to work reliably but feel free to try it.


The only problem with these two is if you accidentally play music from outside the Home Sharing or Downloaded sections of the library.

Block streaming traffic on your router

A final suggestion, depending on what router and/or other equipment for the satellite internet connection you have: is there a setting you can make to block or throttle streaming audio traffic, and/or iTunes downloads? I don't know what servers, protocols or ports you would need to block but you might be able to get that answered with another question here.