The first point is that your computer has an issue.
This happens. You may have a number of identical laptops, with the same hardware and the same software, yet only one of them suffer a certain error. Perhaps one of the components is slightly bad, it may only surface under a set of circumstances not shared by everyone, it could have been damaged when shipped to you...
Here we have (I expect) a problem clearly described, registered in whatever ticketing system is used by tech, notes that you were in the office twice to fix it, etc.
So it's clear that the problem exists. In fact, it's quite obvious you wouldn't otherwise have gone to the office if there wasn't a problem, even if nobody else is (apparently) suffering it (yet).
Second, you have asked IT to fix it
This is the right approach, since the tech department would be the ones that need to ensure the devices work correctly. You have asked if they can change your computer, and that's ok (you can offer a suggestion), and they refused to do so (which is their right to do, as well).
It makes sense that they would prefer to repair the computer. First of all, in case it wasn't clear, you need to understand that changing your laptop would mean replacing it with an equivalent one, if not the very exact same model. A different hardware may fix the problem, but it could also persist (the whole batch might be bad, or if they install a bad version of the drivers in their image, a different laptop won't help).
Tech may think that changing your laptop won't help. They may not have other laptops available. Or they may just want to ensure they find the cause and completely fix the error so it won't affect anyone else in the company in the future.
They might even be using you as a guinea pig to test a new model of laptops. A new hire could be more appropriate for 'experimentation' than an existing user.
Third, you need to travel to the office
Since tech needs you to bring the laptop to the office, you bring it to the office. Obviously, since this involves traveling from your designated place of work (your home) to another location (the office) for business reasons, this shall be done while you are on the clock.‡ If you are expected to work 8 hours per day and your company considers a good use of such time to have you spend 4 of those going back and forth to the office, it is up to them. Not your problem.
‡ You may be entitled to some additional compensation as well, such as a mileage compensation (policies vary from company to company), but I would abstain from asking explicitly about that for now.
Four, you keep your manager in the loop
This is the more important thing you need to do. Keep your manager aware of the problems and how acute they are. Talk with your manager about this if you haven't done so. Send him a message whenever you are asked again to go to the office "to fix the computer they had supposedly fixed the last week". Send another one when "you are finally back after spending 5 hours so that tech could look your laptop and you are able to get back to the work your manager requested from you". I would suggest including as well a little summary of what happened, like if "they claim it is now fixed for good".
It's not your problem if the company has you dancing back and forth to the office. Tech may actually prefer that you go to the office as many times as needed (and they certainly prefer that you travel to the office, rather than sending one tech to your home -spending 4 hours traveling- so you can start working immediately after it gets fixed, instead of spending two hours traveling back).
On the other hand, your manager won't be amused by this. Your manager will to decide how much to wait for tech to fix this on their own, how much to pressure them to solve your laptop problem (either with this one or another) and which is the appropriate way to do that.
You only need to ensure that your manager is well informed of all of this.
Your reputation won't be ruined at all. It's not your fault at all if you were given a faulty laptop or that tech didn't fix it the first time.
Plus, in my experience it is completely normal that any new hire faces issues the first days. Sometimes a piece of hardware won't work, others they don't have access to certain places, a program wasn't installed for them, or even got the wrong Operating System.
Finally, some people in the comments are suggesting you should further break the laptop. Don't do that. There's a well documented problem with the laptop. That's the one that should be investigated and fixed by tech. If you break your laptop they will figure out, and you will be (rightly) blamed for that. That would be a completely wrong move for you.