Do there exist an amenable Beurling algebra that is neither Arens regular nor strongly Arens irregular? In his memoir "The second duals of Beurling algebras", A. T. Lau proved that there exists a weight $\omega$ on $\mathbb Z$ such that $\ell^1(\mathbb Z,\omega)$ is neither Arens regular nor strongly Arens irregular, but this algebra is not amenable.
It seems to be underappreciated by too many authors (this remark is not directed at the OP, but based on my frustrating experiences in several conferences and reading several papers) that what the older results of Gronbaek and White imply is the following: if $A=L^1(G,\omega)$ is amenable then $G$ is amenable and the weight $\omega$ is equivalent to a trivial weight, which means that $A$ is isomorphic as a Banach algebra to $L^1(G)$ (and $G$ must be amenable).
Since $\ell^1(G)$ is AR when $G$ is finite (for trivial reasons) while $L^1(G)$ is always SAI for $G$ infinite (I believe this is N. J. Young's result?) the answer to your original question is negative: no such Beurling algebra exists.