Is it correct to use “But” at the start of a sentence like this?
You can afford it. But think twice before buying it.
Won’t using “But” instead of “However” mark me as a non-native speaker?
If anything it would mark you as a native speaker, because native and non-native speakers alike are taught not to do it, but native speakers grow up hearing everyone else doing so, and ignore those teachers.
It’s quite likely that the condemnation of starting with a conjunction is because some children have a habit of overusing such beginnings:
And then I went to the park. And my friend Paul was there. But he had to go home for his tea. And then I was lonely so I went for a walk. And then I found a toy someone had left behind.
Such overuse is certainly not good, and banning children from ever starting a sentence with a conjunction is easier than telling them to be moderate with it, absolute prohibitions being easier to judge after all. (Though if as an adult writer you find that you are starting a very large number of sentences with conjunctions, then banning yourself from doing so might be a good exercise!)
Of course, as adult writers we should be better able to judge what is wrong with the example I gave than just proscribing all such use.
Interestingly, while you speak of however as another possibility, it is starting a sentence with however that is the more controversial. One common style-guide only changed to permit it in the last few years (alas I forget which) and many still recommend but over however in those cases where one could reasonably use either. (There was an editor of the New Yorker who was forever changing initial but into however, but he was an exception, and even he changed his ways).
Now, I’ll disagree with those people and say that however is just as good as but at the start of a sentence too, but it’s worth noting that the opinion does exist.
(It’s also worth noting, that if you start with a conjunctive however you must followed by a comma while but would not. Other senses of however have different requirements regarding commas, see https://english.stackexchange.com/a/104877/15770 for more).