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Regarding the actions of Saturn, Saturn is where we are tested and disciplined, and therefore may feel inadequate in. Therefore, looking at your natal Saturn, this means you are disciplined towards the affairs that are ruled by Betelgeuse. As another example, Saturn in Libra means you take seriously relationships, and perhaps you might find them to be harder to maintain, or like they had been difficult in your life, thus requiring you to spend a lot of effort mastering them.
I think Betelgeuse interacting with your natal Saturn would therefore be good, as its energies impact Saturn, however it can be hard to tell, because Saturn "prefers" to be in signs like Capricorn which are broad and managerial, whereas it is actually in detriment in the sign of Leo, which is faster and more personal.
Now, that is different than if you raised the energies of the Sun or Betelgeuse to benefit your Saturn, as you would affirm the process to happen positively, which would avoid a negative result or manifestation, like which may exist when Saturn is found within those other signs. It might make more sense to use the energies of Betelgeuse for its traditional functions, like empowering something or manifesting victory and prosperity.
To benefit the affairs of Saturn along the traditional lines, you normally use the energies it is associated with, like Nauthiz, the Shani mantra, or a Saturn Square.
Regarding a Saturn Square or Shani mantra, this means you are inviting additional contractile energies into your life to help materialize or stabilize things, which can invite some degree of stress, so be careful when doing so. Nauthiz is generally viewed as safer.