Links to YouTube video (audio only)
Frankly, it's anybody's guess. Having said that, it appears that from Stanzas 1 and 2 a boy and a girl have had a romp or good time in the countryside under the moon. They probably had sex.
Then, Stanza 3:
Guess I need to run, take me to the bus
But don't let me go, no, don't let me on
Gotta hold me tight, won't put up a fight
Of course I'll stay, I'll stay the night
This mentions the girl asking the boy to bring her to the bus station because she "needs to run," which means she needs to leave. But then she asks the boy not to let her on the bus. Therefore they spend more time together, doing who knows what, at the bus station, specifically at the lost and found, which can be an isolated place after daylight hours (in the US at least). Whatever the case, the rather seedy location of a bus stop lost and found has become a place to remember for the boy and girl. Plus the phrase rhymes with the preceding line.
As far as your suggestions regarding the meaning of spending time at the lost and found, yes, any and all of them are possible. plus the ideas of losing oneself and finding oneself in the love of another and all kinds of other saccharine silly Ellie stuff. Maybe she lost her virginity there and found herself or her true love.
You would have to ask the song writer and or performer and maybe she couldn't even tell you.
And don't wait for the video of this song because music videos rarely match the meaning of the lyrics as song.
But to me it includes celebrating a night together, and that happened to be at the lost and found, and so that will now always be remembered. When they get married and then separated, poor Ellie will always have the lost and found.
So yeah, this question will probably be closed for being opinion based. And that is what "silly love songs" do: allow listeners to make of them what they will, and read into them their own love stories.