Does the state of necessity apply to one who has no option for a definitive cure for asthma other than taking an herbal remedy on which a name other than Allah’s has been mentioned?

If the medicine is purely herbal and free from forbidden substances and impurities, there is no harm in consuming it, even if it was made by Hindus and they mentioned the names of their deities on it, for the following reasons:

The default ruling for herbs is permissibility unless there is a valid Sharia proof that forbids them. There is nothing in Islamic law that prohibits food due to polytheistic invocations. The phrase "that which has been dedicated to other than Allah" in the Quran refers to slaughtered animals, not all foods. Plants are permissible by default, and mentioning a name other than Allah's does not affect their permissibility. Their deities are false and do not affect the original permissibility. There are narrations and sayings from scholars affirming the permissibility of consuming food from non-Muslims and people of other religions (besides the People of the Book) that is not slaughtered.

For this medicine to be permissible for consumption, the following conditions apply: it must be free from forbidden substances and impurities, its benefit and effectiveness must be confirmed, and there must be a need for it to treat an illness or alleviate its harm. A Muslim should pronounce the name of Allah over his medicine and avoid polytheistic invocations.