2 1/8 overhang might be too much for wood as well, as i assume that overhang is parallel to the grain. solid wood isn't all that difficult to snap along the grain lines under the right circumstances
for the reinforcing strips i would not use screws as they can pull pieces out of alignment if one is not having a good day. the adhesive will be doing the work anyway. instead, use this scenario to get a pneumatic brad nailer
i have gotten a lot of mileage out of this one from hazard frought
https://www.harborfreight.com/18-gauge-2-in-1-air-nailerstapler-68019.html
and just about any little compressor will run it but because i'm already on the HF website, this wold work fine
https://www.harborfreight.com/3-gallon-13-hp-110-psi-oil-free-hotdog-air-compressor-57572.html
goop up your reinforcements, put them in place and drive a few brads in. it will go through whatever is in the way and happens so stinking fast nothing has much chance to get out of alignment. it will also go through your finger so be careful
caveats:
do a couple test pieces as close to the actual scenario as possible to make sure the brad head is below the final surface. if they stick out proud of the surface they're next to impossible to set with a nail set. a brad is a small square nail, square nails hold a lot tighter than round nails which works against you if you're trying to set them after the fact.
the glue will encourage your pieces to slip around because it's slippery. i might even go for an Instant Grab adhesive like this, it's white this is just an example. i'd try to get Clear if available, in your case so any oopsies are less visible
https://www.lowes.com/pd/LOCTITE-Po...PD0f7SqiajFsgEqNIZsaAnm2EALw_wcB&gclsrc=aw.ds
"instant" glue + brads= happy. IMO etc etc
if you want belt+suspenders+warm blanket, go back the next day with some screws.