There is no such thing as "custody", the terms are as other posters have said, "residency"
i.e. where the children live. All this is defined by the piece of legislation governing child matters - the Children Act (1989)
As you and your ex were married, he has automatic Parental Responsibility.
As there are no current residence orders in place - you are both equal parents. There appears to be a shared care arrangement in place whereby the children (how old are the kids btw?) are spending a substantial period of time at both parents. There is a shared care arrangement that presumably has been mutually agreed (otherwise it wouldn't be in operation!), how long has the current parental schedule been going?
IMO as a father who has been through the lottery and hell that is the family court system, self-represented myself and got a shared residence order for my daughter (despite the ex trying to go for a sole res order), you're ex has little chance of getting a sole residence order.
It would be a substantial change to the current status quo of the parental schedule. What benefit is there to the kids of your ex (or yourself for that matter) of having a sole residency order?
IMO you should be suggesting either no order - because, believe me, you don't want to go through the family court system, it will damage your relationship with the ex probably irreconceivably, will cost you a fortune financially and emotionally. If you can keep it out of court the better. Suggest going to Family Mediation (google for your nearest family mediation services) to sort out your parenting issues and then get any agreement written up by a solicitor and signed. This will give you some sort of stability.
Suggest it to the ex by letter. If he doesn't want to play ball suggest that you will bring it to court and let the court decide but you will be applying for a Shared Residence Order (SRO)
All a SRO means is that the kids have two homes of equal value i.e. a mums and a dads - which is what they have atm, spending approximately equal time at both parents. Shared residence does NOT have to mean 50:50 so don't acept any crap a tapeworm, sorry I mean solicitor, says to contradict this. There can be a contact element to the SRO which states the division of time at both parents homes.
A SRO recognises the equality of both parents.
Maybe you should think about joining an organisation like Families Need Fathers (FNF), these are a gender neutral national charity that promote shared parenting. They are for PARENTS not just dads, despite the name. See if there is a local branch meeting near to you and pop along for advice