I would recommend rephrasing this sentence. Right now, "should've had" is expressing an unfulfilled past obligation rather than allowing someone else to do something. The "should've had" is a past modal verb, which expresses a feeling of regret that you did or did not do something. To rephrase: I let her fall asleep on her own. I let her fall asleep by herself. I let her fall asleep standing up. Notice how I use the word "let" instead of "had." This is now a basic causative form. You are now allowing her to fall asleep without taking direct action yourself. We can combine the past modal and the causative to create: I should have let her fall asleep standing up. I hope this helps you!

Thank you! So even native speakers wouldn't use the verb "have" this way.

I think this sentence is confusing to an English speaker. Is the sentence referring to a child? If it refers to a baby or small child, one might say "I should have let her fall asleep on her own."