Taking a Lulav in the Sukkah

Sorry for driving the Rabbi crazy with my unrelated questions 🙂
Assuming that there is no sukkah in the synagogue, many rabbis wrote that everyone should take a bath in the sukkah at home before prayer to earn the virtue of the sukkah.
In the vision of Ovadia, he presented a counterargument, since he had before him the commandment of the lulav, prayer, and the recitation of the Shema, and since frequent prayer and frequent recitation precede the lulav.
Apart from the fact that prayer is a rabbinic practice and a lulav is a Torah practice (at least on the first day), I thought I would offer a counter-argument: today, prayer is at a fixed time in a synagogue, so in fact, if I take a lulav before prayer, I am not bringing it before prayer because technically its time has not yet come. (There are rabbis who have even permitted performing a ritual before prayer if it is at a fixed time because it is considered that its time has not yet come.)
A. Does the vision statement actually have any followers?
on. Is there anything real in my answer?
third. According to the Michael-Abrahamic method, we are probably both wrong. If that is the case, I would be happy for the Rabbi to suggest what he thinks.
If I speak nonsense and the rabbi rejects me, I will accept the evil of the decree with love.
(Thank you to the rabbi for all his blessed work, I thirstily drink in his words)