Can you ask her to join the conversation where you talk to the funeral organisers? Face to face interaction is more likely to result in the issue being resolved without unsaid things being assumed on both sides..
SIL, would you come with me to help me arrange the funeral?
At the funeral directors:
You: I want this to happen as quickly as possible so the impact on my children can be minimised.. what dates do you have?
Funeral director: X, Y and Z...
You: Do any of these work for you SIL?
It is likely that one of these will... if not...
OK - what else do can you offer..
M,N,P....
Do those work, SIL? yes / no..
if no, you could say, ... oh, that will be more than a month away, that feels like a long time - can either of you make any other dates work? SIL - how would you feel about joining remotely - it's an honest question because right now, we feel we are in limbo, tip toeing around each other, while we wait to have DH's funeral... it's like we can't really speak, take action or breathe until we have completed this step..
If she won't budge, then you are forced into choosing an earlier funeral without her, or a delaying it for up to 4 weeks... You will step into that knowing the possible impact on your relationship with her and wider family members, going forward.
Or - you email her and say - dear SIL - this is a really difficult time for us, the funeral director has offered us three dates in the next three weeks and all three of them clash with your work dates, it is unbearable for us at the moment and we are really struggling - is there any way you can move one of your commitments or that BIL can help? We dearly want you to be there - it is very important to us, equally, it is hurting us all, the kids especially, to be in this awful limbo - I know that having the funeral won't bring him back and it is an important step for us.