> So I am wondering whether string theory can mimic asymptotic safety.

it's worth noting that all developments in string theory converge to the observation that fundamentally gravity is not a local field theory, while the claim of asymptotic is just the contrary, that it is a local field theory at arbitrary high energy.

Within perturbative string theory itself, gravity is carried by the closed string sector, which fundamentally is not a local field theory, but a string field theory, non-local at about the string scale. 

Withing non-perturbative string theory, gravity is argued to be holographically dual to a local and conformal field theory (AdS/CFT), and as such is not itself a local field theory.

But also apart from the a priori assumption that quanta of gravity a carried by strings, people see the scaling behaviour of Bekenstein-Hawking entropy and conclude that if this entropy is to be carried by quantum degrees of freedom, then this cannot be those of of a local field theory. The AdS/CFT correspondence is in a way a grand generalization of this phenomenon.

A quick, clean, informative  and concise account of this is here:

* Assaf Shomer, "A pedagogical explanation for the non-renormalizability of gravity" ([arXiv:0709.3555](https://arxiv.org/abs/0709.3555))

see in particular the last little section IV.


You might also enjoy googling for *"Jacques Distler" "Asymptotic safety"* to see what one particular string theorist has to say in detail about the AS claims.