I've been sort of turning this one over in my head for a while now. It's a hard question to answer because, as I'm not a law expert, I can't really interpret it from that angle ... That being said, it was certainly possible for men to change their names.

The most famous and very relevant example to look at is John Lyon and Mary Eleanor Bowes, who married in 1766. Lyon was the handsome but penniless Earl of Strathmore; Bowes was a beautiful teenaged heiress - can I make it any more obvious? Their marriage united his social position with her riches, saving an impoverished noble name from ruin. But this was not a fairy tale: the widowed Mrs. Bowes was critical of the match, withholding her consent until her daughter made it very clear that she would have the Earl or nobody, and then the lawyers took more than a year to work out the details. (As I discussed in this previous answer about 18th-19th century inheritance, English marriages of this period involved complex contracts that essentially mapped out the finances of the spouses for the rest of their lives, and after.)

The major issues that the lawyers from both families wrangled over were balancing the tradition of women's property belonging to their husbands with the need for her to be protected in case he misused it, and Lyon's surname. Bowes's father's will stipulated that the man who married his daughter and took his fortune would change his name to Bowes as well, and their children would carry it on. This was controversial because it flipped the script and effectively made Lyon the wife according to tradition, losing his birth identity to be subsumed into his wife's. However, the money won out, and he became John Bowes, his children to be called Bowes-Lyon. (Yep, those Bowes-Lyons.)

So, Simon could certainly have changed his surname to Bridgerton in order to keep from passing on his father's name. However, he would probably have faced some internal conflict over the change in their gender roles, not to mention possibly pressure from other men of his acquaintance, who might rib him about being feminized for taking his wife's surname. Given that he couldn't even bring himself to explain to Daphne, let alone other men, that he refused to have children, I think the conflict over this would have been a compelling and believable driver of the first season of the show. But it might have alienated viewers as too patriarchal and misogynist from a romantic lead.