I've had requests to offer a lower rate through a few different sources of business (besides Rover) and usually it's been for 2 or more dogs. It hasn't happened frequently, but there are a lot of people with serious financial and medical issues (often these go together). It's so personal to share all that life threatening fight, I'd doubt she's trying to scam you.
However, you do have to consider what you can offer and be satisfied with. I'd also consider calendar booking / or anticipated booking volume, and if this may be ongoing booking after the holiday period. Since these are for drop ins, you might still be able to handle other clients, but you also may need to calculate the costs of your travel to ensure the rate is fair. You have to decide what you consider fair first. Then, you can present your offer and get her reaction. I prefer discussing money matters in person, usually at the meet&greet.
I just read your update. That's a total of 8 walks (4 days x twice). Her desired budget of $100 = $12.50/visit and your regular drop in rate is $24. If this was her first booking on Rover ever, she can use a promo code to save $. If you care for her dogs, it may restrict you from caring for many others because you'll spend nearly 40 minutes round trip just driving.
I think that the advice you took to contact her to suggest considering other options and letting her know that you may be available closer to that time based on your availability then sounds like the best move. I've cared for dogs that were 20-30 minutes away 2-4 times per day. It can be very wearing and that drive time plus time spent on walk/food/care definitely restricts boarding clients you can accept. Also, you'll have a fair amount of mileage expense. If you end up helping her out, maybe you can ask if she could budget $140 (which would be 17.50 /visit- $5 more than what she really wanted but $6.50 less than your non-holiday regular rate).