I mean it's not like any prospective wife hasn't had ample opportunity to try to deepen a relationship with him prior, and vice versa.
Meanwhile it's always an option to fess up and clarify it was done to protect Sera who didn't have anywhere else to go.(Not ideal, but people could understand)
And even if divorce tends to be looked down upon more in rural areas(where in more populated areas it is more accepted/common), it's still an option to claim later on if Sera makes it home.
The author shouldn't rush the marriage like you suggest. There is little reason to do so. Nobody in the country is going to ask to see the registry, and it's extremely doubtful there would be any issues with the government in such a small town.
The only reason for marriage, would be if Sera decides to stay, where she could gain legal rights as his wife.
It would be one thing, if this was a extremely traditional society a hundred years back or more which two unmarried people living together would be a massive scandal(effectively crime). Or a society where the local lord could just claim whatever unmarried women he liked(thus Sera would be vulnerable), or women were mostly considered property of men, and had little individual rights.
In a situation like that, actual marriage would be necessary to protect her.(see Highlander series for an example of this)
For Seras situation, claiming to be his wife gives her credibility and goodwill in the community(through Jin having credibility and goodwill), avoids many questions she couldn't answer or the appearance of scandal, and is effectively not any different from actually being married, unless she gets in legal troubles(which would be unlikely). Even trying to get legally married, might carry more risk to it as it could raise questions about where Sera came from(illegal entry of country), or why Sera doesn't have any ID/passport.(Sera doesn't appear to have the attributes to pass as a native, or perhaps believably be added on a family registry or something).
Since Jin and Sera(so far at least) have little reason to legally get married(especially after barely knowing each other), it is better for them to grow closer first. For Sera to make the choice after completely finding no hope or chance of going home, or someone managing to find her. Maybe they fall in love(ideally), or maybe Jin is a good guy and marries her to help her have a secure permanent life in this world(love follows later perhaps).
It would be far easier if Sera can go home, for Jin to just tell people "Actually she was just a guest who needed a place to stay, I lied about her being my wife"(or lie and claim divorce), than it would be to annul a legal marriage with Sera gone.