Generally, EU law (including the rules on the free movement of persons) only regulates situations involving several member states (e.g. a citizen from one member state working in another member state). But the situation of a British national in the UK is a purely internal situation and EU law does not apply, which is why the UK is free to impose extra requirements and you cannot apply for an EEA family permit. Per the official site (my emphasis):
You can apply for an EEA family permit to accompany your family or partner to the UK (or join them after they’ve arrived) if:
- the person you’ll be accompanying or joining is from the European Economic Area (EEA) but not the UK
However, the EU Court of Justice (EUCJ) weakened this principle in several cases (Singh, Carpenter, Zambrano…) and there are ways to invoke EU law even in your husband's country of origin. The most famous one has been called the “Surinder Singh route”. If you qualify for that, you could apply for an EEA family permit and then work in the UK too. See also When returning to the UK from working in the EU, what do you need to show you activated your "Treaty Rights" WRT non-EU family members?
Still, if your husband becomes a Polish citizen, it would not necessarily solve your problems. The EUCJ treated a related situation in the McCarthy case and it ruled against Mrs. McCarthy (but the decision is rather unclear and the details might make a difference).
Incidentally, I don't think that showing the Polish passport on entry makes any difference (not sure there would even be a record of that). What matters is his citizenship and whether or not he can invoke the EU right to free movement of persons.