I think you can take some steps to (attempt to) improve your situation.
in our daily meetings I have nothing to contribute because I rarely make progress on the project
I would address this first, and I would start by examining whether this is really true.
Are the tasks assigned to you too big to be done in one day? Break them down into smaller tasks.
If you spend several hours per day working on this project, you should be able to set realistic goals for each day of work and consequently report on the progress/completion of these goals.
You could for example say something like:
"Today I was working on the expense report submission. I had to change the text on the submit button but the new text did not fit in the available space. I decided to reduce the font size. Since the font size is set in each file individually I had to inspect all of the files in the project to see whether any of them might affect the font size on this expense form. I did get the font size reduced today and the submit button has the proper text now. Tomorrow I will look into the other buttons."
This is not as glamorous as saying you finished the entire form but is much better than having nothing to report.
Once you can verbalize what you are working on and are going to work on next, you open the door for other people involved with the project to potentially provide assistance. One assumes the company is interested in project completion, thus if you say you are going to need a day to figure out this font size issue and your boss knows how to do it in 10 minutes, they might help you out.
Alternatively you may find that nobody knows more than you do (or, said differently, you are the most knowledgeable person on this project). In this case the company wouldn't rationally be able to get the project completed faster by putting another person on it, which means you can stress less/work more normal hours and get your mental health under control.
The client is getting impatient and roll-out is looming.
Is this project developed for the client or in collaboration with the client?
If it's the former, it sounds like your company perhaps overestimated its ability to deliver this project, or the cost to do so. In which case this is not your problem (but your bosses would of course push you to deliver as quickly as possible). Your bosses know that they made a mistake somewhere. Don't call them out on it but work a reasonable amount of time every day, and report what you are doing.
If it's the latter, perhaps the client can provide some assistance/information that would help you get the project completed, if you can articulate what assistance you need.