I have a 2004 SRX with V6. I do not have the towing
package. I own a 380 lb Aluma motorcycle trailer and
a 2004 HD ultra classic about 850 lbs. Do I need a
transmission cooler to tow this MC and can I have one
put on the SRX???

If you do a search, you will find plenty of past discussion on SRX towing and options on how to add a hitch/wiring.

The Cadillac hitch fits great, and comes with the wiring hookup.