You are correct in that the embalming surgeon of the CW period used instruments that would have been commonly used by the hospital surgeon. This practice continues today in modern funeral service.
Then as today, their shared instruments were a scalpel, perhaps also a bistouri, a grooved director (for opening an artery), an aneurysm needle, small dressing forceps, a needle holder, suturing needles, scissors, and dissecting forceps.
Perhaps uniquely, the embalming surgeon employed a hand pump fitted with a stopcock, tubing, and cannulas in various diameters that could be selected and fitted to the pump depending upon the size of the artery being injected. I'm not aware of a period medical application for a pump such as this.
The original pump that I've examined is brass and about the size of a pump to inflate a football. It was a custom product made with an English maker's label, and it along with three ivory cannula and a length of silk-wrapped rubber hose were fitted inside a presentation box similar to the surgeon's medical kits that Ed Archer and others use to encase their instrument sets.
I remain as ever, your faithful & obedient servant,
aka Benjamin Franklin Lyford, M.D.
Drs. Chamberlain & Lyford, Principal Embalming Surgeons
Washington City, D. C.
Adservio mortuus quidnam es non potens adservio ipsum
Traveling with while in the field:
Mid-States Living History Association, Indianapolis, IN
10th Illinois Cavalry Regiment, Springfield, IL
The Society of Civil War Surgeons