A wrist strap is more of a precaution I've worked with people that have been doing repairs for 20 years and never use one and just ground themselves by touching the case. Your hdd drive bays should be un painted or inside of the case clip to one of those. also using a wrist strap doesent give you protection to touch everything however you want. make sure to keep away from the gold contacts on ram and on your mobo same goes for the cpu and gpu. just hold and seat things using the corners ie. the green part. other than that you should be fine dont stress.
edit: So youre right every single thing is painted lol. just hook it to a corner, you should be fine.