is the attempt to bring the term in Cantor Normalform valid, especially how would you rate the step: $$(2^{\omega + 1}+1)^\omega = ((2^{\omega+1}) 2)^\omega$$
The term is $((\omega + 1) + (\omega + 1))^\omega = (\omega2 +1)^{\omega}=(2^{\omega+1}+1)^{\omega}=((2^{\omega+1}) 2)^\omega = (2^{\omega+2})^\omega=2^{(\omega2)\omega)}$
The claimed equalities are true, but not particularly clear imo. I would rather write everything in base $\omega$ and bound the expression:
$$\omega^\omega\le(\omega\cdot2+1)^\omega\le(\omega^2)^\omega=\omega^{2\cdot\omega}=\omega^\omega$$