Exert Ability Workings with Untap Abilities
Yes, you can Exert a permanent multiple times in the same turn like this. The rules on Exert very helpfully clarify this:

701.37. Exert

701.37a To exert a permanent, you choose to have it not untap during your next untap step.
701.37b A permanent can be exerted even if it’s not tapped or has already been exerted in a turn. If you exert a permanent more than once before your next untap step, each effect causing it not to untap expires during the same untap step.
(and a couple of other lines)


Very conveniently for us, all the Exerts only apply to the very next untap step. We won't have any Exerts &quot;lagging behind&quot;, they all wear off simultaneously.