Lowballing breaking point in a car? Only when you can't justify and explain why the offer is that low. But really, there isn't a low ball breaking point. Sometimes, you never know unless you ask.
Also, I hope that wasn't the first car you test drove.
Things to do when you are looking to buy a pre-owned car that more then a couple years old. You should do this with any used vehicle but the older it is the greater the need to check. First and foremost, go to wither carfax.com or autochecker.com and look it up by VIN number. Yah, its going to cost you a few bucks, at about $20 for unlimited uses for 30 or 60 days or roughly that.
Then check out the vehicle history. Things I would check for is how many owners. Although the car you are looking at has a few years and miles, I really shouldn't have more then 3 or 4 owners previously if even that many. Anytime the vehicle changes registration this is always up dated. Also check for problems with major accidents, and look into any minor ones also reported.
If that all checks out, take it to areliable 3rd party mechanic. Explain to the mechanic ahead of time that you are merely checking out a used car before you purchase it. Try to use a mechanic you know, but if you can't, use a mechanic NOT in the immediate area of the used car's location. This is just incase the used car dealer has made goody goody with a local mechanic or two next to him/her.
After that, find out your budget, what you really ARE willing to pay. Make sure your state tax laws. I know in Texas I still pay full sales tax on cars sold between private parties. It's the only item I know that does this and is collected at the DMV upon registration. Your state may have similar laws and that may change the price your willing to pay by a lot. In my state that turns a $7000 vehicle into a $7540 vehicle after taxes. Then there is registration, and inspection and/or emissions if they aren't up to date. A $700 vehicle can turn into a $8000 vehicle in a hurry. Not to mention you might need repairs and parts for the car soon for a used vehicle. You already know this since you say the car needs new tires soon.
Personally, after all that, I know you already checked KBB, but also check Edmunds.com and NADA for relative value of a used car. You will notice that all 3 will have different but similar values for the car, usually a few hundred dollars apart in difference.
Then, take the LOWEST TRADE IN VALUE you find fom all three, subtract any work costs that might need to be done on the vehicle and take off another 3% to 5% and make that your starting offer if you still want to go through with the vehicle purchase. Make your way up to the LOWEST private party trade in price minus the cost of any immediate work, like changing out the tires, that needs to be done. If they won't accept, then move on. If they question the offer you came up with, merely ask them if they want to come with you to check it out on the internet. Take them to the site that quoted you the LOWEST for private party or trade in values. You can't go wrong with this methodology when buying a used vehicle.
CLIFFS
1) check vehicle out on carfax or autochecker
2) test drive more cars and check those out
3) Look up your state tax, registration, inspection, and emission laws.
4) find values for all cars you are interested in on KBB, Edmunds, and NADA. Use the lowest values you find for all 3.
5) Start slightly below trade in value and work your way up. If you don't get a taker, move on to the next vehicle. Be sure to point that out to the seller as a last ditch effort.
5) Buy the vehicle you want, at the price you want, and enjoy it.
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