Starter removal

It should come out but if not remove the through bolt from the right side motor mount and jack the engine up a couple of inches.

Here is guide to help you see what you are in for when doing the job.

https://www.2carpros.com/articles/how-to-replace-a-starter-motor

Here are some diagrams of what it is like on your car (below)

You will need to raise the car as well

https://www.2carpros.com/articles/jack-up-and-lift-your-car-safely

Please let us know if you need anything else to get the problem fixed.

Cheers, Ken

Sounds like the starter solenoid or solenoid linkage is worn. Your best bet is to change the entire starter and solenoid as a unit. You could just change one or the other but, it'll come back to haunt you if you do. Disconnect the neg term on the battery, then it's just 2 bolts (15mm) that hold it in place. NOTE; If there are any metal shims between the old starter and the engine block, be sure you reuse them on the new one. Hope this helps.

Is it staying engaged after the engine starts?

Each casting is not 100% identicle. Sounds like a shimming problem, but I would expect the starter to be screaming when you crank it. Is the mounting surface clean or surface corrosion present? Did the gear loked lined up well? Is there damage on the flywheel teeth? The flywheel may have had a weld let go from the original drive gear problem if it went out of aline. Try to get a real good look at the flywheel if possible.

Check for DTC po 338 in the computer first.

Replace the crankshaft position sensor if this codes is present. Check engine block at starter boss, the starter drive housing, and the engine flywheel starter ring gear for damage from this condition as well.

I don't know what it is that you think you're bypassing by you can't use a push button. The vehicle has an immobilizer.