Doing it is a bit complicated, but not terribly so.
You'd need DoubleShadows
X3 Mod Manager and Editor to start with.
Next, you need to find the latest version of TShips. If you're running a Mod (selected from start screen) that alters/adds ships, it'll be in the 'mods' folder.
If not, it will USUALLY be file 09.cat/dat in the game root directory. Do note, that if you are using a mod that opporates as a false patch like XTM, the last patch file (usually 11.cat/dat) will have the necessary TShips file.
Now, run the X3 Editor, select 'TXT Editor', and open the appropriate mod/false patch file, and find the 'types\TShips.xxx' file and select it. When it opens, you'll be editing an in-memory copy of TShips.txt.
In the window on the left, you'll see a list of ships that are in the game.
There are actually TWO entries for Mobile Pirate Base. They are, however, exactly the same, but it's difficult to pick which in the game SE without help, so you need to change the class for both just to be sure.
The easiest way to find BOTH entries, is to switch the list to 'ID' mode, then check the 'Sorted' box. This will sort the list by the ship ID. Scroll down till you find the entries 'SS_SH_TL_P' and 'SS_SH_TL_P2'.
Left click on one. The display on the right will switch to the first page of the ships stats. The item you need to change is 'Class'. Currently, it will show 'TL (8)'.
Double click on the class, and a pop up will open. Click the arrow to the right of the selection window, and the dropdown of eligibal classes will roll out. The currently sellected TL class will be at the top. Scroll down till you see the other TL entry, then select it. Hit OK
The Class of the ship should now read 'TL (0)'. This is the same class as all the other race TL's use.
Change the other entry as well.
Click the Save button at the top, click save in the dialog, and overwrite.
If all goes well, things won't crash. Once you run the game, the MPB's should now be able to DOCK and drop stations like normal TL's.
I don't recommend changing the TShips in the official 09.cat/dat files, if that's your only choice. You can use Mod Manager to extract the file, make a new patch file (10.cat/dat usually) and then add the file (make sure the path is types). Just in case something screws up and you don't bugger you game.
Alternately, you can make a copy of the 09.cat/dat files, just in case.