Hmm, everlasting? Kind of hard to say.
The Fendi peekaboo has been with Fendi since 1965.
https://www.purseblog.com/fendi/handbag-history-the-fendi-peekaboo/
It is very much intertwined in Fendi's history and in terms of longevity it is still here today.
However, it's not the most "popular" of bags and it's resell value drops 30-50% the moment one buys it and walks out of the store.
(Reminds me of Lady Diors, classic to the house, but also depreciates heavily.)
The andiamo bag is a new bag for Bottega. However, it does have that classic intrecciato weave that also has a long-standing history with the brand. Only time will tell how "classic" this particular bag will or will not become.
To be honest, when I voted, I didn't read your entire post and judged based solely on appearance/design. I did consider the peekaboo at one point but retail vs pre-loved price always held me back and I eventually lost interest.
I prefer the andiamo bag, not because it's newer, but because I love the intrecciato weave (I have a Bottega roma) and frankly I think it looks more interesting and more "three dimentional", than the peekaboo.
However, I have read concerns about the potential design flaws of andiamo and the wear of the leather rope as it gets pulled through the grommets repeatedly. If one goes with the small size and doesn't put a lot inside, I don't forsee that being too much of an issue.
If I got the andiamo, I would not wear it crossbody. That rope is thin and would dig into my shoulders. The peekaboo would be more comfortable crossbody, especially if you bought a wider strap (since they do sell straps).
Anyway, just my two cents. In general, I think both have very classic shapes and designs, so you can't really go wrong with either.
Good luck and happy shopping! Please let us know what you end up deciding on!