= Solution
Use the single-edge <heat-bath Markov chain>: choose an edge uniformly and resample it from its conditional random-cluster law. If its endpoints are already connected without that edge, its conditional open probability is $p$; otherwise opening it merges two components and the probability is
$$
\frac p{p+2(1-p)}=\frac p{2-p}.
$$
Detailed balance makes $\phi_{p,2}$ stationary. Driving this chain and Bernoulli heat-bath chains by the same update edges and uniforms gives the stochastic domination
$$
\operatorname{Ber}\left(\frac p{2-p}\right)^{\otimes E}
\preceq\phi_{p,2}\preceq
\operatorname{Ber}(p)^{\otimes E}.
$$
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