Gears Not Meshing Properly

check how true the axle is on each gear...
some bad sets (not SHS) have had a pi$$ed axle
set at like 88 degrees instead of 90 so the gear itself runs buckled

examine each tooth at back of sector & inner ring of spur for chipped teeth
or a bad burr

check sector on its own - if shimmed too high below the sector
the tappet lug can rub against the shell on the top
check this, if the cam/lug rubs you may need to mod/sand the box
(if you can't alter the shimming to respace the sector)

you have narrowed it down a little to those two gears perhaps
now see if you identify the exact issue - likely to be one of those gears

In extremely rare cases a 12:1 spur could have been supplied with a 13:1 sector
the spur inner teeth on the 12 will be larger
and the sector on 13:1 will have a larger ring of teeth
these together may not be compatible as they will try to push the gears apart

Could have a moody set
or
could be the way the box/bushings are set they just slightly bottom out on the teeth

I have noticed on some boxes and gear sets the bevel gear can sometimes sound rough - sometimes swapping out a stock or similar 10 tooth bevel as most gear sets still use regular 10 tooth bevel gears
On noisy sets, I remove the bevel and ensure spur+sector run smooth then try another bevel
In your case you have done this but the spur+sector is the issue

bad chipped/burr on teeth or bent/pi$$ed axle comes to mind
check the allen key bolts are done up correctly and axle not crooked

EDIT: afterthought....
SHS gears are very chunky and often the spur gear if set very low
can rub on both bevel & sector bushings underneath
This doesn't happen on smaller 6mm bushings
kind of very very lightly scrapes on 7mm bushings
but on larger 8mm & deffo on 9mm bushings they will rub if spur gear is set too low on deep bushings

actually think this is cut off lever rubbing on bushing if I'm honest
the gear rubbing produces and opposite arc or convex/concave point of wear
but you get the idea that a deep bushing on bevel/sector can rub on the outside edges of spur

check see how low the spur gear is set underneath & if gears run true & spaced ok

Hmmm, I'd probably not tbh - depending how rough/smooth they stil run
you could "run them in" but only by hand spinning them
rough running under power/stress will promote wear in bearing/bushings

TBH - I'd order either a new set maybe try a ZCI/Core set of 13:1's
personally I have rated SHS over Core/ZCI as the bevel isn't great
but you probably could use the SHS bevel if it runs well/better

What is your old gear set ratio ?
I had an old VFC box that had quick 16:1 gears in
doesn't the spur & sector have a spring on them?
anyway to quickly find your gear ratio.....

chuck all 3 gears in the open box
set the sector's first tooth at say 12 o'clock
mark with pencil the bevel at 12 o'clock
rotate the sector gear carefully 1 full complete revolution
counting the bevel turns as you go....

multiply the bevel turns by 3 to get you gear ratio

eg: 6.25 bevel turns = 18.75 which is the actual ratio of 18:1 gears

I had a vfc set which came out at 5.5 = 16.5:1
most 16:1 sets are 5.75 = 17.25:1 ratio so vfc set was a touch faster

12:1's work out at 4.25 = 12.75:1 ratio

your old set might not be that slow is what I mean so you could stick with those

or try another 13:1 set and see if that runs smoother
possibly might be able to mix & match the two 13:1 sets
BUT this should not be done unless you are still having grief
and only with paying very very close attention to how compatible they are

You should not really mix sets of gears as a rule of thumb
but bevels often can be used if they mesh/run ok
and 18 & 16:1 sectors are usually the same & compatibleif you check carefully
hence if you get another decent 13:1 set the spur/sector have a chance
you might find you need to experiment or mix/match

could of got a defective or cheapo snide fake set of gears
I got a couple of fake "pink" shs m4 nozzles that was wrong length and just didn't fit on tappet or cylinder head correctly
You could order another shs set from elsewhere to avoid same issue from similar batch perhaps

Sorry I can't offer more help, I'd say try the shs gears in another box if you have one lying around & if still no joy contact the seller/retailer
Sucks but bad batches, fakes and just simply some stuff just works great in some builds and other stuff in some builds just don't seem to run so sweet as expected