Nisaa
Yes. Istihadah does not close the mosque to a woman the way hayd or Janabah do, and nifas follows the same restriction as hayd for as long as it lasts. She does not treat bleeding outside her known cycle as menstruation in the first place, and that difference is what keeps the mosque open to her.
The evidence that keeps a menstruating or junub person out of the mosque, [An-Nisa' 4:43] together with the hadith of Umm 'Atiyyah (Radi'Allahu Anha) (Narrated by al-Bukhari and Muslim), is addressed to hayd and Janabah specifically, states of major ritual impurity. Istihadah is neither. It is treated as a continuous minor hadath, comparable to chronic incontinence, the same description that lets her pray and fast without a pause. None of the evidence that keeps hayd or Janabah out of the mosque was ever addressed to a state like hers.
There is also direct evidence that she may remain inside the mosque, not merely pass through it. 'Aishah (Radi'Allahu Anha) narrated: "A woman among the wives of the Messenger of Allah (Salla Allahu 'Alayhi Wa Sallam) performed i'tikaf with him while she had istihadah. She would see blood and yellowish discharge, and a basin was placed beneath her while she prayed." (Narrated by al-Bukhari). I'tikaf means dwelling inside the mosque, day and night, for an extended stretch, not stepping through it once. If a Companion of the Prophet did that with his knowledge while bleeding from istihadah, entering to pray a single Salah is not seriously in question.
What istihadah still asks of her here
What istihadah still asks of her here is the same precaution that governs her prayer elsewhere: a pad or cloth to hold the blood back, the routine described for her in the hadith of Hamnah bint Jahsh (Radi'Allahu Anha) (Narrated by Abu Dawud and al-Tirmidhi). The basin placed beneath the Companion woman above is that same precaution, carried further for a stay measured in days rather than minutes. What istihadah asks of her in the mosque is protecting the space from her bleeding, not staying out of it.
This sits alongside the rest of what istihadah leaves untouched: her prayer, her fasting, tawaf, and touching the Mushaf. Bleeding from an IUD is one of the more common ways this comes up in practice; see whether it stops her prayer for the same reasoning applied there. If what you are bleeding might actually be hayd rather than istihadah, telling the two apart first is what decides whether any of this, including the mosque, applies to you at all.
Hayd is the opposite case entirely: it interrupts an i'tikaf outright rather than merely asking for a precaution; see what if your period starts during i'tikaf for what leaving the mosque means for the days already spent there.
