A165 coupe bucks like a bull on launch

Hi, can tell you’re a bit anxious to get to the conclusion, but I need some more Info.to help you out. No worries.
With No idea of the Car’s year and exact Model, what Year and condition of newly installed
Parts required for your swap (if any were rebuilt/ known to be Sound-or used a donor Car/Truck?). I’m Not sure- i’ll guess you modded a 1965-72’ Mustang, OE Manual, 200CI or 250CI Straight 6 w/an 8” Rear.
Maybe upgraded to a Top loader 4 Speed or a (T-5) 5 Speed, upgrading to an 8.8” or 9” Posi Rear end, a rebuilt 289.(?) Make sure the Bellhousing Bolts are Tight to the Motor/Trans.
Weld in any reinforcements/Frame Connectors in the rear for Strength? It’s irrelevant if the guess/speculation is correct, once you elaborate -I’ll be happy to help you to work out your “launch” issue(s). We’ve all needed a hand, been there.
Does this occur when your Rear’s break loose- upon a launch, and a whole Vehicle “Chatter” and loud sound(s) start occurring- Steel to Steel, a rhythmic and rapid “Thumping” noise- sounds off like the Car’s gonna fall apart? Does it seem not to occur while you’re already cruising and you get on it- hard, or occur- but differently when the Car “plants” and takes off W/no wheel spin?
If the last information I’d guessed on is close, i’d say It’s “Wheel Hop”, a fairly common occurrence where the Leafs on both sides both actually “Twist”, as both Tires try to ‘grab’ the road. the Rear end fights to stays in place, once the Spring twist becomes so tight- this Torsional load exceeds that of the Tire to Road Traction- resulting in the entire Rear end Torque suddenly unloading. It Recoils (twisting the Leaf Springs back to level, this causes the Suspension (And Tires) to “Hop” when it occurs-Making lots of noise...
It quickly does this many times- within in a single burnout or during a hard launch- everything Vibrates, bumps into the Car’s Floor, Exhaust hits the Axle, etc. unpleasant- yet not unusual or difficult to remedy.
With the older muscle and Leaf Springs-it’s frequently an issue, even newer Mustangs with Coils Spring Rears with improper (Junk) Shocks/Dampening Rates-this may occur. Once we solidify what your setup is, I’ll show you a few simple inexpensive ways to eliminate it. Really Sounds much worse than it is.
I’ll see if you’ve uploaded any new info. later tonight/tomorrow. It’s all good, simple fix. J

Welcome.
Fluid level will not affect a stick in this manner.
When you say "launch" do you mean just taking off from a stop or launch as when you are racing and popping the clutch?
If it is only when taking off in first gear from a stop but is fine when you shift to second and other gears then that is sign of a bad clutch which is slipping and causing hots spots on your flywheel. If you installed a new clutch but didn't have the flywheel resurfaced then this can cause it too because once those hot spots get on the flywheel the only way to get rid of them is to have a machine shop resurface the flywheel. If you used an old pressure plate it will also have hot spots on it and needs replaced.
If this doesn't answer your questions then more info is needed.

Really don’t have enough info to answer this one yet...

OP you are experiencing a thing known as clutch chatter. it can happen for many reasons including, but not limited to;

poor driving technique
bad engine mounts
bad clutch surfaces
bad trans mount
bad clutch linkage
bad pressure plate
bad clutch disc
bad pilot bearing
bad throw out bearing

these are the most common issues that can cause clutch chatter.

if you rushed through the engine swap, you might have a combination of all of the above.