Show that the set of all TMs that move only to the right and loop for some input is decidable
Your approach is on the right track. It should follow the next ideas:


If there exists such $w$, then, when $M$ reaches the first "blank" after reading $w$ it will be in a state $\tilde q$ that goes right (and then keep going right, no matter the state changes). Finding this $\tilde q$ that satisfied this property (if such exists) can easily be verified for by checking all the states in $Q$ (of the given $M$).
If such $\tilde q$ exists, we need to see if it is reachable after reading some input $w \in \Sigma^*$. The idea you already noticed is that if this happen, then $\tilde q$ will be reachable after processing the end of some word in $\Sigma^{Q+1}$ (or the like). This is simply a pigeonhole argument..


Therefore it is enough to only examine all the words in $\Sigma^{Q+1}$ and check if $M$ on any of them goes only right, and reaches $\tilde q$ when hitting the first blank. Since the number of words in $\Sigma^{Q+1}$ is finite, this is doable in a finite time.

You just need to properly formalize the above ideas as a decider $\hat M$.