I dont think there are any stoves that can work with those clearances. It certainly is not safe as a fireplace either. And the vinyl tile would need to go to.
@bholler has helped me and countless others here many times and is an extraordinary support and expert, however, I'm not sure if I agree with this. I dealt with close clearances to a mantel (albeit not as close as yours) and was stumped for years! I finally resolved it two times, and I believe both would work for you:
1) First, I put in the very small but handsome Jotul F100 and used top and side mantel shields which I custom made with 22 gauge steel, a pair of metal cutting snips and some high temp paint. I met clearances and passed inspection no problem - I believe you would too. Here's a link to the manual:
http://www.tallpinesfarm.com/web_ext/Owner's Manual/139678_R01_Manual F 100 NA_EPA.pdf. If you are rear venting the stove (which you should so you don't have to stuff the stove in the firebox loosing a lot of heat to your brick), you will likely need double wall flue pipe to connect to your liner, which has a 6" clearance if you get the right stuff (I use Selkirk). I soon realized that the F100 was waaaaay undersized for my space, and the short burn times and barely any room to get 16 inch splits in drove me nuts, so I went back to the drawing board (see number 2 below).
2) Second, I discovered that the people at Woodstock not only make some of the world's greatest wood stoves, but they will also help you every step of the way, including looking at photos and measurements of your setup, and your whole home size and layout, as well as what you are trying to achieve (i.e. 24/7 burning, just supplemental heat, etc.), and will help you every step of the way and stand behind you and their products. If you are willing to pull a free standing stove out a bit in front of your mantel, you can have up to 24 inches of vertical (with a slight incline upward) double wall stove pipe from the back of your stove to your T connecting to your chimney liner. You will almost definitely need far less than the 24 inches and will likely just need to get the stove a few inches in front of the mantel, and thus will make top and side clearances (Woodstock will discuss if you need to take rear clearances into consideration doing this, and if so, their stoves have rear heat shields to reduce clearances). I believe that this can work for you, and you can get one of the best stoves out there with a few different models to choose from.
A few other considerations you will need to take into account are what shape are your chimneys in and how big they are on the inside, as you will likely need to put in a full six inch insulated liner. In terms of your hearth, you can easily put a simple hearth pad down on top of the existing vinyl, which you can build or purchase, and will need to take into consideration if you need just ember protection or a certain R value, which will depend on the stove you go with. This is something you can purchase or make. I decided to go simple, easy and relatively inexpensive with this:
https://www.amazon.com/gp/product/B01MSAG5ZP/?tag=hearthamazon-20 and it actually worked out quite well. It's not a beautiful work of art that some people have for their hearth pads, but it isn't ugly either and is simple enough that keeps the focus on the stove.
Tell us a bit about your space, layout and square footage, and what you are trying to achieve with heating and we may be able to be of more help. Also, start cutting and stacking your wood asap, as you'll need some time to get wood dry enough to burn (depending on the wood, if you get some now, you might be ready for next season).
Good luck and keep asking questions!
