They told me not to bypass the amp because either the stereo wouldn't work or it will heat up too much.
That's a load of crap! I installed an HU in my brother's car two years ago now. It does not overheat. Hell mine hasn't overheated yet! They are making up excuses because they took a shortcut and got caught. The factory amp will make your Sony deck sound like a$$. If you get the amp bypassed it will sound much better.
As long as the stereo has power it will work. There are kits made to bypass the amp. It will heat up more now since the on board amp is on but that's normal (amps generate heat) You should go back and tear that idiot a new one for lying to you IMO.