You can definitely say
under the delusion
M-W defines delusion and gives an example with under:
something that is falsely or delusively believed or propagated
- under the delusion that they will finish on schedule
As for deception, it is rarely used with under,and mostly without the article:
Looking at the some contexts, I see that being under some deception is common in religious texts and refers to being deceived by something or someone rather than deceiving oneself.
I would stick with under the delusion. Of course, under the impression is a set phrase but has milder connotations.
to be under the delusion of something false is tautological. Also, both your sentences do not make sense grammatically. I suggest beginning them with Being under or simply Under and adding a comma after idea and promise respectively. I am not very happy even with this result, but I have already crossed the line of proofreading, so I will stop here.