I'm not sure why you think "let" might be a stative verb (let is "causative" but not stative as far as I know).
"stative verb" means the verb describes a "state" rather than an action, like the verb "be" in most of its usage.
But you should go by the meaning in the given context rather than thinking whether a verb is stative or not.
As far as using such verbs with "would" for past habitual actions, here are some examples:
- I would be a lawyer in those days, ... (the lawyer job is a state, not an action, so it sounds funny)
- I would be a shy child, ... (same - "a shy child" implies a personality, not an action)
But sometimes "be" can be used with such "would" too.
- I would be deliberately funny but you wouldn't get it.
"be" here means "acting in a certain way", so the phrase means "I would act funny but you didn't understand my jokes".
The point is that the intended meaning is the important thing, not just which verb is used.