If they want to pay $1,300/mo as is, saving you the Reno cost.

I would need to consult tax consultant for this however I expect to reduce my effective costs by ~$1,495/month (based on a 32% tax bracket).2.

Ignore the $1,250 per month because that's your cost of housing, if you didn't pay that on the home you'd have paid that in rent so that's a non factor. 

Here are some indicators to help you assess a buyer's seriousness:Questions: Serious buyers will ask detailed questions about the property, such as maintenance costs, neighborhood amenities, and property taxesStrong Offer: If a buyer makes a strong offer quickly, it often indicates a high level of interest and intent to purchasePrivate Buyers: If approached by a private buyer, ensure they provide a Letter of Intent (LOI) and proof of funds.  

Purchasing a “distressed” property and “repositioning” it requires significant capital for improvements, build outs, holding costs, maintenance.In looking for that “needle in a haystack” slam dunk great deal you’re competing with very knowledgable, greatly experienced investors with the cash to make no contingency offers and close in a few days.  

(It’s multi-family but not entirely stabilized.)Rehab Costs: How much should I budget for bringing vacant units to market-ready condition?

I will say, short term rentals require a bit more work, and if you are moving out of state, you may want to take a look into getting a property manager ( take that cost into account when running the numbers ).I hope that helps, reach out if you have any more questions. 

Should I go through with taking ownership of the property and try to salvage it for profit or do I refrain from even involving myself in the situation as it may cost more than it's worth to save it?

About the same as if you rent it, less costs.

After my P.M. costs I collect $940 or so/month.