@FilippoAlbertoEdoardo Just got a copy of this book and indeed, you are right. The group he calls $\mathcal{T}_p^{\mathrm{ord}}$ is what I called $\Gamma$. In Chapter III, Remarks 2.6.5, Gras gives a formula for the order of $\Gamma$ in terms of the $p$-adic regulator of $K$ and other simple arithmetic quantities (assuming Leopoldt, so this order is finite). I think in the example I described, $\Gamma$ is cyclic, so knowing the order is the exponent. Thanks for the help.

Yes. See my edit for one case.