I want to offer my condolences. I'm not sure if I can help you with your specific question, but I would like to share my own experience about loss and bereavement leave. When my mom died in 2013, I was naive and didn't get bereavement leave or ask for it. Instead, I used sick days and worked remotely (even from inside the funeral home). In hindsight, this wasn't a good decision on my or my employer's part, but I was so eager to please my employer that I put my job before myself and my family.

If you don't want to straight up ask for extra bereavement leave days, you might phrase it like "I'd like to take bereavement leave and return next Wednesday" (request it in writing), where returning next Wednesday would net you a couple extra days. If your employer wants you to use your PTO and only gives you three days of bereavement, you may want to request that you be able to use your vacation days instead of sick days.

I wish at least that my employer hadn't made me use up most of my sick days for a parent's funeral. In my case, there were a couple days down the road where I had strong memories of my mom and wanted to take a day off, but couldn't because I had used up all five of my sick days. What made things difficult for me was that I started to have stronger feelings about her death over six months after the event, which can be difficult for an employer to understand. Your employer might be completely compassionate and understanding now, but when the grieving process moves into a different stage, they may not accept what's going on. They might also not understand that you may need time for your children down the road.

Employers can be downright selfish and sinister. I'm only speaking in generalities and don't know or judge yours, but even with traumatic events like this, an employer can become frustrated with your time off. I wish I had done a better job being assertive with my employer about getting time off right away and not using up sick days. Make sure you keep your own records, and if you ever do need to see a doctor for depression or anxiety, make sure you get a doctor's note and keep a record of it. When I did have to use a sick day and had run out, my employer not only lied to me at the time and fired me, but fought my unemployment through the adjudication process as well - and lied again and again. I had records of my recent doctor visit and documentation of communications which vindicated me in the end. It was a blessing in disguise because now I'm at a much better place.

Don't be afraid to assert yourself in getting time away from work. I wish you the best and hope you heal well.