You can add any kind of fat that will encapsulate the capsaicin from the chili. This can be:
- Dairy (full-fat milk or cream)
- Rendered fat (along with creme or butter)
You can also add a sweetener to offset the impact of the capsaicin, but that's not ideal for biryani. With the fat, what happens is, the 'hot' parts of the chili get coated and sort of disappear into the dish. An acid alone is likely to just make the dish too spicy and tart.
I would add a little more vegetable oil and some heavy cream with butter to tone it down. Add it a little at a time. Then use some citrus (lime, calamansi, etc) along with some additional ground cardamom to bring out the sharper notes again. You can add more chicken stock to thin it out. But here, the acid is just to put some flavor back, not really hide the chili.
Coconut cream/milk will also work, not quite as well as full dairy cream.
Note, make small adjustments then wait for 5 - 10 minutes to taste, or you'll easily take out too much flavor with the cream.
Use a fat to absorb some of the capsaicin from the chili, but do it in a way that won't make your dish too oily.
This will alter the taste of your dish, it can also hide notes like cardamom and clove and cinnamon/anise / etc. You have to put that flavor back in, along with the fruity taste of the chili.
Since serving over rice, you can add a little more stock just before spooning it over to help tone it down some more (as a final resort)