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Indeed, the example of a ball (see e.g. Iosif Pinelis's answer below) shows that $d_{\Omega}$ should rather be concave in this case. Anyhow, the answers below seem to take this into account.
The examples the OP is interested in are a very special kind of non-reductive Lie group - to wit, the semidirect product of a semisimple Lie group with a non-compact Abelian Lie group. Can such issues be circumvented in this case?
As far as I know, the (three-dimensional in the OP) Poincaré and Euclidean groups are not reductive, so I have no idea. That's also why I insist that you're actually asking for a notion of $G$-Higgs bundles for $G$ non-reductive - the case of $G$ reductive but not semisimple is already covered by the definition presented in the OP and the examples you're interested in are (I think) non-reductive. You should amend the title and body of your question in order to address that, they are slightly misleading as they stand.
Don't you mean to ask for a notion of a $G$-Higgs bundle for $G$ non-reductive instead of non-semisimple, since this was your initial hypothesis on $G$? There are (real) reductive Lie groups that are not semisimple (e.g. $G=GL(n,\mathbb{R})$) and your definition applies equally well to them.
Hmm... According to what seems to be the state of the art in understanding the problem (called the "unbalanced Procrustes problem" in the numerical analysis literature), it certainly appears so - check e.g. L. Eldén, H. Park, A Procrustes Problem on the Stiefel Manifold, Numer. Math. 82 (1999) 599-619 and Z. Zhang, Y. Qiu, K. Du, Conditions for Optimal Solutions of Unbalanced Procrustes Problem on Stiefel Manifold, J. Comput. Math. 25 (2007) 661-671. See also math.stackexchange.com/questions/4492668/…