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 safity 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.
Within 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)
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.