I few things to address first. The county appraisal means next to nothing in most cases, so don't use this figure toward any calculations. I also wouldn't be particularly concerned with offer structure at this point either unless you intend to propose some form of seller financing, which is going to be secondary to the number you propose anyway. Also, be very careful about what you think you resell the house for, what the rehab will cost and what would like to buy the house for.
When analyzing scenarios such as this you really need to reverse engineer them by identifying the most accurate comps (bring in an agent if you struggle with this but just look for properties of similar size, characteristics and location that have sold in the last 6 or 12 months). Once you have that number (maybe it ends up being $160k, maybe it doesn't), take 70% of it (or maybe even 65% if you're new to the process), then subtract your renovation, holding and closing costs, which will vary based on a number of factors.
In your case, let's play it safe and say the the home ultimately sells for $150k. From there, 65% of that figure brings us to $97.5k. We then subtract $40k for reno (because it will more than likely run on the high end of your estimate), bringing us to $57.5k, which is below the initial offer they received from a realtor and since you don't want to work your butt off for $5k, you ultimately need to offer them less than every other offer they've already turned down.
Long story short, really focus in on your estimates throughout the equation and prepare to move on when you can't make a competitive offer. Don't start moving the goalposts (i.e. by inflating the sell price or lowering the reno budget without data to support it; using 80% instead of 65-70% in the equation above to sacrifice your eventual return) in an effort to wedge a square peg through a round hole.
Not every property you analyze works, and providing them with a "low" offer isn't taking advantage of them. As an investor, that is simply the amount you can provide in order to get the desired return on your time and energy. If you were looking at it as your primary residence, could get it for $100k and put $40k into it and instantly have $20k in added equity, great, but that's not the direction you're approaching it from. When looking at a home for an investment purpose, you're offer is generally going to be "low" or else it just becomes a really expensive hobby.