I'd put the home on 'Hold Do Not Show'. Days on market do not accrue when on 'Hold Do Not Show' status in my MLS, so perhaps they don't in yours, either. Or maybe you keep it active with an explanation.
If I were you, I'd write and drop or door knock a letter stating that your pre-approved or cash buyer (whatever is accurate) wants to buy a home in that neighborhood. State that they are prepared to pay market value for the right house (if they are) and write some of the items on their wants/needs list. Also say that a showing does not obligate the potential seller to sell their home.
State that if the homeowner already has an agent, they can have the agent contact you to arrange for a showing. This adds authenticity to your letter. Potential sellers are less likely to think you're trolling for listings for an imaginary buyer.