“Grammatical sense” is not the idea I think; there is a question of grammar only when a construction is used (verb, subject, complement).

Seeing “temps” anywhere as a standalone word, someone thoroughly familiar with French would probably think that what was meant was either the concept “time as the fourth physical dimension” or the concept “weather”; those concepts are not usually made precise by using a complement; this is not the case for the grammatical concept; for this one the term “temps grammatical” is sometimes used to make clear what’s meant. To be unambiguous and direct in your message you’d have to lengthen it a little; for instance something as “(Le) temps qui passe” would do; unfortunately there seems to be nothing shorter. For instance “temps passant” does not stand alone as well.

The shorter “temps fuyant” refers unmistakably to time as the physical quantity but it’s not neutral in its reference to time; it connotes the rapid passing of time, it’s also an idea about time that poets will readily exploit and a mingling with poetry is possibly not ideal in the make-up of the message.

French:

Je trouverais cela bizarre, je penserais à un tatouage inachevé et je demanderais éventuellement quel est le reste de la phrase ou du proverbe.

Le concept évoque aussi pour moi les inscriptions sur les cadrans solaires et anciennes horloges, mais je trouverais bizarre également de voir du français plutôt que du latin : Tempus fugit ; Vulnerant omnes, ultima necat, etc.

En tout cas, j’associe spontanément le mot temps sur un tatouage à time et pas à weather ni à tense.

English:

I’d find it weird, it would make me think it’s an unfinished tattoo and I might ask what’s the rest of the phrase or proverb.

The concept would also remind me of inscriptions on sundials or ancient clocks, but it would be weird to see them in French and not Latin: Tempus fugit ; Vulnerant omnes, ultima necat, etc.

Anyway, I spontaneously associate the word temps with time, not weather or tense.

To me it sounds (or looks ?) a little weird.

It can work well with other standalone words though. For instance, there’s a character in Overwatch with a “Cauchemar” tattoo. It fits, it makes sense. It sounds good when you read it.

But “temps” as a standalone is different. I think it’s because it’s polysemic (as you said temps can also mean “weather”, but it also means “tense” (the verb form)), and it’s also very homophonic : there’s tant, tend, t’en, taon.

Or maybe it’s because as a one-syllable one it just doesn’t sound that good when said alone?

I’m not 100% sure that’s the reason why it feels off. But maybe a very small sentence with the word could work better.