"Is it not the stance of the Bhagawad Gita (and consequently the Vedas)
that matter only exists because we perceive them so?"
No. Matter is not just a perception.
BG 13.20 — Know that Prakruti and Purusha, both are certainly beginningless; Transformations and Guna (3 modes, sattva, rajas, tamas) are known to be possible due to Prakruti
If what you called "Matter" is "Prakruti", then Bhagavad Gita accepts it to be permanent (i.e. beginningless & hence endless). It might be retaining itself forever by continuously changing in cycles forever. Mostly, Prakruti is also represented by asat (BG 2.16).
"Which means that matter does not exist without being perceived."
If you perceive anything, it exists, but not otherwise practically. e.g. We may not know what exists at 10 feet 30 degree angle. Because we are indifferent towards it. Similarly, certain people are indifferent to everything else, except the true self (Atman). For them, relatively matter doesn't exist as they have attained their true nature.
"A Matter is a matter of interest!" :)
"So if the permanence of matter is asserted, shouldn't it imply the permanence of consciousness in the material world which perceives it?"
Yes, it implies permanence.
We saw above, both Prakruti and Purusha are forever. Some people term Purusha as "enjoyer" (or set of "enjoyers") and some as "consciousness" (or set of "consciousnesses").
Analogically, think of a river (Prakruti) and a swimmer (Purusha). Swimmer jumps into the river and takes the form of many fishes (Jeevas). All different fishes will perceive river in various ways. Towards the end, all the fishes will magically merge back to the swimmer. After taking rest for a while, the swimmer will jump again. This keeps repeating.
"Which implies permanence of earthly life."
Not necessarily for everyone.
Continuing above analogy, certain fishes would assume the swimmer's original form during beginning itself, certain at midway of river, certain at 70% completion and certain at very last. So depending on a kind of jeeva we are, we shall get the peace when we assume our true self.
Earth is a too localized platform, but probably you meant "state of continuous engagement in Prakruti". Well, that means there is no Moksha, but there is certainly a state where a person doesn't have to take rebirth in any possible form. So Moksha does exist.
Now that's a debatable topic, if the Moksha is permanent or not. To me as of now, Moksha isn't retained beyond 1 Kalpa (Brahma's day/cycle). After that everything repeats again. If that interests you then, you may refer below post:
What will happen when all atmans get moksha?