If you have the resources and the discipline to be sure that you can pay that loan off before interest is charged, the no-interest credit card is the better way to go.
Normally, you should not put a charge on a credit card which you cannot afford to pay for with cash equivalent at the time of purchase, because the amount with interest that you'll likely wind up paying is much higher. This post has some good advice.
However, there is generally value in aligning your expenses with the timing in which you will receive the benefit of those expenses, especially when there is not an interest cost. For example, if someone offered me a no-interest 25-year loan with regular monthly payments on solar panels guaranteed to last and perform for that long or longer, I would take that even if I had more than enough in the bank to cover the cost up front.
Sometimes, life events happen that require quick liquidity of assets, with expensive consequences if you don't have that flexibility available. By having this medical expense on a no-interest loan and keeping the savings secured in the bank, you are buying time to find creative solutions to deal with those situations should they come up, and in this case you're getting that for pretty much free thanks to the credit card promotion.
Will this affect your credit? Somewhat, but temporarily, especially if you manage it well (in which case it'll help you long term). If you're planning to apply for a mortgage (or other large interest-bearing loan) while the account would still be new and the balance would still be high, especially if you don't have a lot of available credit to keep overall revolving loan utilization low, this might be a bad idea and cost you more in the long run, particularly if your credit score is close to some threshold that would lock in different interest rates for a longer-term loan. But if you'll have it paid off before applying for that big loan, you're likely better off having proven you can manage credit well instead of having tapped savings.
As far as other fees, read the terms of the card carefully. Use of balance transfer / account access checks often incurs a fee of ~2% of the amount of the transaction, and if you're signing up for a new card advertising "no annual fee" be sure that's a permanent thing and not just "no annual fee for the first year, then $495 per year thereafter" or something like that. Usually this sort of fine print is not too long and those terms are not too hard to find as long as you're looking for them.
Also be sure you know what happens if you miss or are late on a payment, and don't do that. Sometimes, if you are even a bit late on one payment, you lose the no-interest promotion and have to pay the full amount of interest at the credit card's penalty rate, sometimes even retroactive to the time the purchase was charged. This is where the discipline of always making your monthly payment on time makes a huge difference.
Note that medical costs are a leading cause of bankruptcy in the US, and if some other unfortunate situation does land you in financial trouble, it is often easier to deal with medical debt as medical debt instead of as credit card debt. Working out a monthly payment plan with the medical provider, including trying to negotiate the overall price to be lower even with that plan, is probably a very useful first strategy, with the credit card option as a backup. It may even be better to forego points bonuses on a credit card to keep the debt as medical instead of credit card and have better financial flexibility with the savings.
Note: This answer on a related question may also be of interest.