If I understand your question correctly, you are asking if $k'\le ck$ for a constant $c\ge1$ implies that $\mathcal{H}_{k'}\subset \mathcal{H}_{k}$. Now Theorem 3.11 in V.I. Paulsen, M. Raghupathi, An Introduction to the Theory of RKHS, Cambidge Univ. Press 2016 says the following: $f\in \mathcal{H}_{k'}$ with norm 1 iff $(x,y)\mapsto k'(x,y) - f(x)\overline{f(y)}$ is a kernel function. I think this implies the claim.  

If your set of inequalities holds for $k'=k'_\omega$ for $\omega $ in a measurable subset of the underlying probability space of probability $\ge 1-\delta$, then on this subset $\mathcal{H}_{k'_\omega} = \mathcal{H}_{k}$. In this sense the equality of RHKS (which might or might not be an event) occurs with probability $\ge 1-\delta$.