Is it permissible for a menstruating woman who has performed the Sa'y and shortened her hair, thereby exiting the state of Ihram, and then performed the Tawaf after becoming pure, to do so in Umrah by analogy with Hajj?

A woman's ihram while menstruating is valid, and her refraining from circumambulating the Kaaba (Tawaf) during menstruation is also valid. However, performing the ritual running (Sa'i) and shortening the hair (Taqsir) before the Tawaf in Umrah is incorrect, because the permissibility of performing Sa'i before Tawaf is specific to Hajj and not Umrah, according to the more preponderant opinion. The release from ihram (tahallul) occurs after Tawaf and Sa'i. You have erred by performing Taqṣir before Tawaf, but there is no expiation (fidyah) upon you due to ignorance of the ruling. You must perform Taqṣir now.

It is preferable and more cautious to return to Mecca to perform Tawaf, then Sa'i, then release from ihram by shortening the hair. If you are unable to do so, then the Umrah is valid, Allah willing.