Firstly there is nothing particularly difficult about rebuilding a mower engine.
It is really nothing more than common sense and a bit of knowledge
You have the common sense & we have the knowledge.
Now as to the electrics
What is on the old engine will swap over to the new engine in about 1 to 2 hours if you have never touched a spanner before in your life
Once the flywheel is off it is nothing more than a couple of screws & a plug .
Lastly
The advice from the dealer was ultra conservative
From the scheduled service fee chart I use to bill with , rounded out & condensed a bit
1) clean mower & engine bay 0.25 hrs ( usually takes longer )
2) remove & replace the engine 1.25 hrs ( usually takes longer )
3) strip engine 2.25 hours ( done in about 1/2 that if the flywheel comes off easy )
4) hone bore .5 hours ( about right , includes cleaning )
5) replace both pistons & con rods 1.0 hrs ( about right , not a job to be rushed )
6) Rebuild engine 1.0 hrs ( about right )
7) run & adjust 0.5 hrs ( usually a bit longer )
You come to 7.25 hours x charge out rate
So for me that is 7,25 x 72 = $ 522 + parts and these are Aust $ not US dollars
a pair of pistons & rods were around $ 300 last time I bought them, gasket kit is $ 45 with seals and if needed a head was $ 225
SO If I was to rebuild your old engine it would be ≈ $1100 ( AUS ) a new engine is $ 2700.
Lots of dealers do not want to open up engines because their techs could have replaced 30 belts in the same time or done 3 full services for twice the profit of rebuilding the engine .
If the replacement engine is cheap enough then toss it in and go mow as you are still in peak mowing season .
Pull the old engine apart take lots of photos .
I very rarely have to replace pistons & rods unless one is broken & your symptoms sound more like blown head gasket which is a $ 200 repair than a worn out piston as they usually throw oil everywhere .