How can one when not affiliated to an institution get access to very common piece of equipment for a subject?
Find someone in the institution (your approach 1), write to them, and ask.  You'll need a brief description of your experiment so the recipient knows what they're agreeing to.  If this is common equipment in your field and you're not proposing to take over the lab for hours every week, just ask.  The worst they can do is to say no.