My intuition tells me that my daughter needs to understand that we still love each other,

No, your daughter needs to understand that you don't love each other anymore but that you both love her enough to want to be respectful to each other.

I want to be able to spend more family time together even though we are separated is this even an option?

You and your ex husband aren't a family anymore and you shouldn't pretend that you are. One of you will inevitably have another committed relationship with someone else and this will be very confusing for your daughter if you've been telling her that you and her dad are a family.