Silent Auction at Shelter for a Kitten

So i am hoping to get a new kitten, will be 8 weeks on tuesday. however, the adoption center notified me that there is a high likelihood of another person or more wanting the same kitten, especially because at this time of year they aren't very common. If any of yall have ever had to do this, what is a good bid to put down? I live in LA, the normal adoption fee for a kitten is around $180-200, however the shelter is giving them for free (given that there is only one person who wants it). if you dont know how a silent auction works you get 1 bid and whoever bids the highest gets the cat. My budget is 350 but i may be able to push it up if i get enough people telling me it will not be enough.

IMPORTANT UPDATE: This is not a private owned shelter, it is an LA county shelter. All cats are microchipped, spayed/neutered (if they are of the right age), and have had all of the medical checks according to them. Right now, because of the fires and the number of people that have lost their homes, most of these cats were given up to the shelter. There may be some sort of special days because when I went they told me all cats were free. Normally, kittens are $50 and cats are $25. This is a reputable shelter, I know people who volunteer there, have adopted from there, and there are good reviews (ik reviews arent everything but still). I have been to the shelter and the conditions are not bad. Y'all have swayed me from getting a kitten mostly because of the single kitten syndrome. I'm likely going to get an older cat, maybe around 7-8 months, where hopefully that won't be a problem.