Should we use "like" as a conjunction?

Can we use it?

Well people do, so we certainly can, at least informally.

Is there any grammatical argument that we can't, leading us to perhaps avoid it in formal use?

No. There's not and like has had this sense since the Middle English period, and never lost it.

Indeed, it's the preposition sense you had fuller confidence in that was once controversial.

Well, why do some people object?

Hard to say. It could be a reaction to it becoming more popular in the middle of the 1800s. This rise in popularity could have given some people the impression that it was a complete novelty and not "proper". Of course that's incorrect, but it can be seen to be the case with quite a few "zombie rules" ("rules" about how to write English that have no actual bearing in grammar, history, convention, or the works of the great writers of the past), someone gets it into their heads that they're defending good old fashioned English and actually brings in a new rule it never had.

It could also be based on objections to like as a filler. This dates from around the same time (but not in all regions where it is now used), and seems to come from its use as a conjunction. Ironically one reason for objecting in addition to objections to fillers generally, is that it misuses it as a conjunction and weakens the statements that it is part of. ("I was, like, really proud" could be parsed as suggesting that I wasn't really proud, just something approximating pride).

Should I avoid it anyway, because I'd rather be seen as correct by all, than know I'm correct in the face of objections from those who hold to this zombie rule?

That's the difficult bit. While the "rule" against using like as a conjunction is not just nonsense, but demonstrably just nonsense, the fact remains that people do object to it. Sadly, this can lead to it reading as unpolished even to people who know better, especially with heavy use.

I personally would advise to consider rephrasing like to as is, but if the given like form scans better to you than the as if form, then keep with that form, and to hell with the snoots!*

*Applied to language, snoot for a soi dissant language-snob comes from David Foster Wallace's essay "Tense Present", which reads well, but has much nonsense. It can be good sport to scan it looking for errors of the sort a reasonable person would forgive, but a self-described snoot would object to, of which there are very many. Of course, that doesn't stop Wallace being a great writer, - and I'm sure he could have found vastly more minor errors in my writing that I in his - but that in itself argues against the thesis of the essay.

There never was a general principle as to why like could not be used conjunctively, and it is now strongly supported by corpus data from around the world.

(‘The Cambridge Guide to English Usage’)