I would always work any offer with the exception being one that is so lowballed as to be intentionally insulting.
6% under list is not insulting.

I would certainly value a real offer more highly than a verbal declaration of interest.

If you are FSBO and you have a cash offer from someone who isn't using a Realtor, aren't you saving money? If the Realtor brings you a buyer, won't you have to pay her some commission? If so, include that in your calculations.

I'd tell the realtor that you have an offer in hand and need to counter by tomorrow (or whatever the deadline is). If the other party is really interested they'll light a fire and bring their contractor back.

In the meantime, I'd prepare a counter offer to the "bird in the hand." I would counter as though there were NOT another potential party (i.e., don't let your "greed" emotions counter your logical decision making). With that counter, you can let their agent know that you have another very interested party that is getting close to making an offer.

And don't let the cost of the fix-ups cloud your judgement on the value of the house. If your realtor's comps were valid (and you do need to look closely at them), then market value is market value.

If you want to keep the offer alive, you have to either accept or counter. By doing nothing, it can go away. Nowadays an offer with no contingencies is worth a lot. The realtor who contacted you may just be fishing for a listing so I wouldn't base my decision on that contact. Keep the lines of communication open with your current buyer and see where it goes.

I would suggest re-evaluating your thought process.

What you want from the house has no bearing on whether a counter-offer is a good idea.

What you've put into the house has no bearing on whether a counter-offer is a good idea other than as a reference to the realtor's "appraisal". (Note, realtors often overestimate housing value to get a listing and then pressure the seller to lower the price to market values, so use caution when relying on realtor's "appraisal").

In your shoes, I would heavily value the no contingency offer, but string along the first offer as long as possible while trying to pressure the realtor to deliver a second offer. Depending on whether or not the second offer materializes, I would counter back aggressively or at 3% below listing as a counteroffer is pretty standard in most real estate negotiations.

Be aware that unless you are offering a commission, the sellers are typically responsible for paying their realtor... unless the realtor cons you into paying.

I would take that offer