In French you wouldn’t, in any case, say “invitation for war”; a proper term would be “avis d’appel sous les drapeaux”; (it might be “notice of call for military service” or “requisition letter to go to war” in English.)

• Mon grand-père a reçu un avis d’appel sous les drapeaux.

Another way to say that, but less literary

• Mon grand-père vient de recevoir une lettre de réquisition pour partir à la guerre.

The expression une invitation à la guerre does exist in French, but it is a soften/unofficial version of déclaration de guerre and is usually not to be taken literally.

Appel sous les drapeaux is idiomatic but doesn’t (necessarily) relate to war, just that you have to serve the army for a period of time.

When this relates to a starting war, the most idiomatic expression is:

Mon grand-père a été mobilisé en xxxx

Regarding the uncertainty expressed in your second sentence, rest assured that using the notion of getting an “invitation to [join the Army/become a soldier/fight a war]” to euphemistically characterize “being drafted” is perfectly okay in English (at least in American English, where the invitation is often described, as it is here under the caption & in paragraph three, as having been issued by/received from “Uncle Sam.”
(from rockdalenewtoncitizen.com)

If you are trying to at least approach this euphemistic tone in French, although “appelé” is used officially (i.e., non-euphemistically) to describe matters of conscription in French (just as “called-up” is in English), “appeler” can include the notion of “invitation” (which is included in your English version) and therefore, I would nevertheless suggest something like the following:

Mon grand-père a été appelé à la
guerre
(paraphrased from Six ans de bonheur, d’angoisse et d’espérance,
by Pierre Martin, via Googlebooks)