Engine powered down going up hill then recovered, PTO stopped, now engine will not turn over

Carb needs adjusting. It is not receiving enough fuel with the extra load being put on it by the pto. Sometimes the carbs are preset for an area with a higher or lower altitude.And most manufacturers just preset it so the engine will start. Then you have to set it for your application. They also sell the same carb for different models and engine sizes.The jet needs to be turned all the way in til it stops then turned 1 1/2 turns counter clockwise. Start the engine engage the pto and slowly turn the high jet screw counter clockwise until the engine smooths out. You could just try it with the screw is set at now and adjust it that way. Let me know if this helps cause I will assist you until we get it corrected.

Newer mowers have a safety interlock that prevents the mower running when going in reverse. My guess is that the reverse switch is on and when the PTO is activated the engine dies like you shut off the key.
Second idea is that the device the PTO drives is frozen up. Test this by trying to see if it turns with the engine off. When you engage the PTO the engine chokes down because it can't turn the PTO.

I had a similar experience and I had to replace the PTO.
What size motor do you have and what brand is it?

check reverse safty switch or engine rpm to low to run pto

Whenever you activate the PTO, it puts an added load on the engine. It drags the RPMs down as the engine overcomes the inertia of the attachment's moving parts: blades, auger, etc. You feel and hear the RPM drop every time. If the engine speed is dragged down too far, it stops. There are several things that can cause this. I'll list the easiest to fix to hardest. Remember, you might have more than one thing wrong. Dirty carb and dull blades might stop the engine when one alone wouldn't.
Attachment problems:
Attachment needs more RPM when you hit the PTO.
Too much resistance to the implement. Mower deck in thick tall grass, snow blower clogged with snow & ice.
Blades need sharpening.
Drive belts too tight, or debris around pulleys. Blades jammed. Belts routed incorrectly after switching attachments. Underside of mower deck clogged with old grass composted over winter
Blade hub bearings need to be lubed. Need grease gun to fit the little metal nipples, called Zerk fittings & a cartridge of "all purpose" grease. BTW you should lube all Zerks regularly to protect bearings from early death.
Diagnose: Disengage PTO, turn off engine, and turn the blades manually. If there is a lot of resistance, the trouble is with the attachment, not with the engine or PTO.
Engine problems:.
Dirty air and/or gas filter; or bad gas. Air intake clogged, Mouse nests under hood. No kidding, happened to me.
Need new spark plug. Old plug can cause power drop even if engine runs.
You may need to go to a pro for more difficult work, such as a bad or broken PTO clutch, compression loss in engine, weak fuel pump, or other engine problems. Your engine is not particular or unique to your tractor. Go to the engine manufacturer's web site for troubleshooting procedures. Also, go to one the many small engine experts here on Fixya. My experience is mostly related to the working machinery.
FINAL POINT relative to all equipment problems. If the easy fixes don't work, check to see if there is any time left on your warranty (if any). If there is, or you don't know, call the manufacturer's customer service department. Even if they only give you some advice, the complaint will be logged and the warranty clock stopped if you need professional work later. Save the case # if they give you one.

The first thing I would do would be check the fluid level in the radiator. Sounds like the car is overheating under load due to lack of coolant. If the coolant level is ok then the radiator cap may be leaking pressure and causing the car to overheat.

which engine? 2L I4 or 2.5L V6?
chocking, it bogs full time.
total loss of power at all times? any speed, any hill, any gear.
is this a A/T or M/T car, the A/T can slip. easy
does the engine lack power, or does the tranny slip.
slipping is easy to discover, the RPM of the engine rise fast
but you go now where fast.
bogging can happen in many modes. (all or just 1: pick one)
all permutations
1: hot engine?
2: cold or both.?
3: wont race even in neutral, but idles ok. at 800 rpm.?
4: only with a fast right foot,called tip in throttle. but recovers.!
5: lost power at all driving loads, and worse on hills.?
6: all the above.?
in a real shop, i drive it and in a few minutes all the above is seen and vastly more, that is why online help is so bad, (me too)
no inputs. = wild guessing.....
heck i even notice things like?
dash volt meter dropping. below 14v at any time running.
overheating
CEL glowing or worse flashing (USA feature , OP never said)
old noises and vibration.
clutch slipping or A/T slipping.
Scan tool connected, and seeing pending codes, or any odd PID behaviors.
bad idle speed, idle speeds not controlled correctly.
and 100 more things, Ill stop here, it's a complex machine.

flashing is usually a sign of a bad misfire situation, check the plugs and wires, also check for coolant in the oil or oil in the coolant.

First. My disclaimer. I do not have first hand experience with your model mower.

Well, it sounds like when you engage the PTO the power to the PTO shorts to ground causing a) your charge light to come on as the voltage drops substantially and b) your mower to stall as the minimum voltage of the ingnition system is reached (or some other effect is at work). The clutch is might be internally shorted, or you may have a frayed wire in the PTO circuit. You may or may not have a PTO relay. If you do, it is sure to burn up if you continue using your mower. If you do not have a relay, you will definitely burn out the PTO switch. It is odd you are not blowing an inline fuse if this is the correct answer. Check your PTO circuit with a continuity tester to tell you if there is a short (or measure resistance directly with a voltmeter on ohms).

Sounds like a loose belt check the tension spring if they ok check and see if belt is strectched