When I do complex two-sided things, my preference is to make a fixture in a piece of sacrificial MDF, then secure the part in it, then cut as a follow-on operation w/o changing the zero or re-homing.
For my last 2 sided project (two sided coasters) I got pretty good lineup by:
In a sacrificial MDF piece, create a pair of dowel holes. this toolpath is centered in my project, and around the axis I want to flip. I cut each side of the coster in one pass (left is the up face, right is the down face).
After cutting the pair of dowel holes I save this xy location in CM using the quick actions.
Create a separate location where you cut the two dowel pin holes into the project. I did this to the side, and saved this location in quick actions as well.
I put my project over the dowels, then screwed the piece into the MDF. After a run I flip and then run the same project again.
Doing it this way was pretty quick, and I was within 1mm after the flip, which was fine for my project. I did this using Carbide Create & Carbide Motion, but there wasn’t anything special in the project to support the flip, other than just being perfectly symmetrical around the flip axis.
Is the geometry based on the center of you stock for both sides? If that is off then when you flip the second side would be off.
I looked at the dowel method but in the couple of 2 sided jobs I have done so far I simply used a square on the spoil board to register the part. Flipped along the Y axis and all was good but that is the way I did the mirroring in CC.
With fusion 360, you can set your zero location to anything that you want, including a hole that you machined in the stock. I use a 3D touch probe to find the hole, but there are lower tech ways you can do it as well which should be pretty accurate. To do so, what I would suggest is:
Setup a fence that is square to the machine movement (e.g. nail down a piece of wood as close to square as you can get it and then run the CNC down the edge to make it very square)
Machine the 1st side however you want, but machine an additional 0.25" hole in the stock
Flip the stock along the fence (aka, keep the same side of the stock on the fence so that the cuts are all oriented relative to the same face of the stock)
Set zero by jogging until a 0.25" endmill or dowel drops into the hole that you machined
Machine the 2nd side in this orientation with the zero set to be the hole center