OK, firstly I'd recommend you don't even consider a gunsmith or, God forbid, a garage mend until you get to know what is possible with the scope and mount. Not trying to teach you to suck eggs and, if you know all this, just ignore but maybe it'll help.....
The base/mount....if it is an LSR.......should look like this:
External:
Internal
There are two parts to both the SSR and later LSR:
1) The base should be fixed to the receiver by pins, screws and locking screws as you can see. There is no (or at least should not be any)adjustment on this part of the assembly. If you have "wiggle room" then most likely you have oversized holes and the mount will simply work lose unless you loctite it to the receiver.
2) The mount should be a three part assembly consisting of the dovetailed lower section and two legs with rings which are dovetailed and screwed into the lower section. This has coarse windage adjustment on the rear leg either made by turning opposing screws to give left/right adjustment or by using a spanner/wrench on a bolt located in the same position as the large screws you can see on the rear leg in the top pic (which mechanism you have will depend on whether it is an original or close copy).
This offers a large range of adjustment and, given the extremely small movements needed at the receiver end of business to have a large effect on target, it is hard to believe that it would not have more than enough to compensate for even the most ham-fisted attempt to mount the base.
If not...shim the front or rear of the base to move left or right. This is not a great fix as it means you won't have the optimal metal-to-metal contact along the length of the mount and receiver wall...but short of machining the mount as you suggest, it may be your next best option.
When making windage adjustments on this mount, make sure the screw beneath the front leg is just loose enough to allow the front ring/leg to rotate or you will end up with problems related to stress/distortion of the scope tube...and remember when you are done adjusting, to retighten.
When I refer to shimming the legs, a gunsmith should be able to insert shims under the legs (front or rear) to move the scope elevation in the necessary direction if you have run out of internal adjustment in the scope.
However...before doing any of this...look at your scope:
ZF39 is the generic model type and was available in Zielvier (x4), Zielsechs (x6) and Zielacht (x8) configurations. Provided yours is an original, the top adjustment was the same on all of them. If it is a repro..check before proceeding.
The scope turret:
There are three adjustments can be made on this scope. From the top down:
1) The rotary bar control is for focus
2) The knurled and numbered thumbwheel is for elevation (with an additional side mounted locking screw to secure) and has a limited range of travel. The actual available travel/adjustment inside the scope is greater than the wheel can access.
3) Internal adjustment and slipping the scale can be accessed by the three screws on top of the turret.
Once you have run out of room (up or down) on the thumbwheel, carefully loosen (BUT DO NOT REMOVE!!) the three top screws. Loosen off the knurled screw that secures the thumbwheel, The elevation thumbwheel should now be disengaged from the internal mechanism allowing it to move freely. Simply move the wheel to the opposite end of the numbered scale, retighten all the screws and you should find you can continue to move the reticle in the required direction. If not, it means you have got to the end of the internal adjustment and then shimming the necessary leg becomes the best option.
A translation of the ZF39 manual is published in the second volume of Richard Law's "Backbone of the Wehrmacht" - beg, borrow (but don't steal) a copy and scan/copy those pages - getting your head around old scopes can be "interesting"
If this rifle were mine and the hacksmith who attempted to do the conversion has screwed it up so badly that using the available mount/scope adjustment or shimming can't fix it, I'd take the mount off and enjoy the rifle for what it is - a 70 year old rifle that shoots well with irons. Come back to getting it scoped when you have the money to get it done properly.
And by "properly" I mean a combination of machining the base to align with the bore and, if also necessary, shimming (or possibly machining?) the legs for vertical alignment to the bore...also, bear in mind that you may need to lap the rings if you shim/machine the legs.
I would not, under any curcumstances, be tempted to start opening more holes in the receiver....EVER! You're pretty much stuck with that and have to work from that baseline.
One thing all my projects has taught me is they always cost more time, money and effort to get right than you'd get back for the finished article were you to ever sell it. And that is using a gunsmith who doesn't get it wrong too often.......
Putting things right from a position of complete cock-up can be like trying to climb Everest in a tu-tu and ballet shoes! Best to accept when you are doomed from the start.....
Good luck!