Translation:The government's budget is high because of the budget of the army.
Why is the Hebrew sentence structure not parallel, with a smichut at the end of the sentence but not at the beginning? In English we would normally say "government's budget ... army's budget" or "government budget ... military budget" or "budget of the government ... budget of the army". Using different structures, while grammatically correct, is considered poor rhetorical style. Is the Hebrew sentence likewise clumsy, or are the rules of style different in Hebrew?
The Hebrew sentence sounds fine to me as a native speaker. Since ״תקציב הממשלה״ is a pretty common term, it could have been equivalently used. I have never heard of any rule against mixing any linguistic structures.
By the way, in general, using smichut is considered better style than not using it (i.e. using "של"), and thus it is more common in writing and in formal settings (e.g. news broadcasts). In everyday language, using smichut may be perceived in many cases as old-fashioned or pompous, as if you're trying to artificially improve your style. That is, it may sound unnatural when using it for non-standard terms.
The use of "budget" twice is OK, and you'd never refer to the army budget as an "allowance". However, the suggested English phrasing is awkward. You're much more likely to hear "government budget" than "government's budget", and "defense budget" or "army budget" rather than "budget of the army".
Sadly, these versions are not accepted. For some reason, DL doesn't seem to accept the English equivalent of smichut (I'm not sure what the correct grammatical term is), even when the Hebrew sentence contains a smichut.
נתניהו מבקש תוספת של מיליארדים לתקציב הביטחון
The phrase "התקציב של הממשלה" is a standard possessive phrase. The phrase "תקציב הצבא" is a סמיכות ("smichut", or construct state), which is a special form of a possessive phrase, that makes the של drop and requires that only the possessor is definite (in this "הצבא"). Equivalently, one could have used "התקציב של הצבא" or "תקציב הממשלה".