Can I tape cement board with mastic instead of mortar?
It doesn't matter.
Waterproofing is behind tape example:

Here you have your waterproofing behind the cement board - old school method.   I would always use thinset because it is easier.  Get a bucket of thinset going and water it down bigtime.
Tape and put just a slight slight layer right in the crack.   You should definitely be able to see tape, but nothing behind tape.
Let sit for 20/30 mins to stabalize, add mix to your bucket and start tiling
I don't see the point of using mastic here because you will be layering thinset right on this anyway.  The tape is just there to join the two pieces.   In reality you could tape and tile at the same time although the logistics of the tape falling would probably triple the time of taping in the first place - unless you have a tile magician.
the reason for &quot;it doesn't matter&quot; answer is that if you use it here there is still thinset over the mastic.

Waterproofing over the tape:

Here you are going to put something like redgard over the tape and cement board.
Materials for tape and bonding have very little importance.
ACTUALLY - the industry is in need for a solution here.  Tape and thinset is terrible as the amount of thinset causes a small ridge which may effect tiling, especially smaller mosaics.  Honestly I would use duck tape with no bonding if you take inspectors out of equation.
So mastic is just fine, just go light on it as you would thinset.   Remember you aren't hiding the mesh tape like drywall tape.

The real answer is you are putting too much thinset on.   Like I said - just enough to hide behind tape, not the tape.   The rest of the thinset that splattered should be scraped with putty knife and then wiped with wet towel immediately.   Big shower is 3 minutes clean up after taping if that.