The answer is given by Valmiki a few Sargas later. In Sarga 46, Sita tells Ravana (who has come in the disguise of a Sanyasi, with a design to cunningly abduct Sita):
samaashvasa muhuurtam tu shakyam vastum iha tvayaa || 3-47-22
aagamiSyati me bhartaa vanyam aadaaya puSkalam |
ruruun godhaan varaahaan ca hatvaa aadaaya amiSaan bahu || 3-47-23
Meaning: Be comfortable for a bit of time, here it is fit for you to stay (temporarily), and my husband will be coming soon, bringing enough forest produce, and on killing stags, mongooses, wild boars he fetches sufficient meat.
i.e. they lived on meat and on whatever produce they could gather in the forest. Even in the Sarga whose last shloka you quoted (sarga 42), Sita wasn't concerned about the life of the deer, i.e. she was not enamored by the beautiful life sprinting there in the form of that deer. Instead, she expressed clearly that if that stag doesn't come "into capture" of Rama willingly, then he should kill it so that she can happily enjoy sitting on that deer skin or decorate her palace walls back in Ayodhya with its skin & antlers. i.e. It is the mesmerizing skin & anlters of that deer, which Sita was after.
And whoever will start about vegetarian Rama & Sita, please learn Sanskrit langiage with grammer first. Then read Valmiki Ramayanam ON YOUR OWN.