Unless I am making a typo, the result is: $$ 2^{2s} \frac{\Gamma(\tfrac n2+s) \Gamma(2s)}{\Gamma(\tfrac n2)^2} {_2F_1}(\tfrac n2+s, 2s, \tfrac n2, -|x|^2), $$ where $n$ is the dimension and ${_2F_1}$ is the Gauss's hypergeometric function. See Table 1 on page 168 in my survey [1], or Corollary 2 in the original paper [2].
References:
[1] Mateusz Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019, https://doi.org/10.1515/9783110571622-007
[2] Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki, Fractional Laplace operator and Meijer G-function, Constructive Approx. 45(3) (2017): 427–448, https://doi.org/10.1007/s00365-016-9336-4
Edit: The hypergeometric function is continuous on $(-\infty, 1)$, and — as described, for example, in this DLMF entry — we have $$ {_2F_1}(a, b, c, -|x|^2) = \frac{p}{|x|^{2a}} \, {_2F_1}(a', b', c', |x|^{-2}) + \frac{q}{|x|^{2b}} \, {_2F_1}(a'', b'', c'', |x|^{-2}) $$ for some (explicit) constants $p, q, a', b', c', a'', b'', c''$. In particular, $$ {_2F_1}(a, b, c, -|x|^2) \sim \frac{p}{|x|^{2a}} + \frac{q}{|x|^{2b}} $$ as $|x| \to \infty$. Therefore, as long as $n > 4s$, we have $$ (-\Delta)^s [(1 + |x|^2)^{-2s}] \sim C_{n,s} |x|^{-4s} $$ as $|x| \to \infty$, for some (explicit) constant $C_{n,s}$. Obviously, $(-\Delta)^s [(1 + |x|^2)^{-2s}]$ is continuous, so we immediatiely get $$ |(-\Delta)^s [(1 + |x|^2)^{-2s}]| \leqslant C_{n,s}' (1 + |x|^2)^{-2s} . $$ In order to get a similar lower bound, one needs to check that $C_{n,s} > 0$ (which seems to be the case when $n > 4s$, as long as I did not make any mistake), and that $(-\Delta)^s [(1 + |x|^2)^{-2s}]$ is never zero. I am no expert here, but another DLMF entry seems to suggest that the total number of zeroes of the hypergeometric function is $$\lfloor s \rfloor + \tfrac12 \operatorname{sign}(\Gamma(-s) \Gamma(2s) \Gamma(\tfrac n2+s) \Gamma(\tfrac n2-2s)) = 0 ,$$ as long as, again, $n > 4s$.
(Double check for typos.)