So you need to be clear on entry as it is clear this is a key issue and pillar of your business to be a safe place for those with allergies.
"welcome to x - just a reminder that no outside food is allowed due to allergies and health and safety issues."
If they raise any external issues such as a severe allergy, diet, etc then they can have the option of a special wrist band with the items allowed and checked written on it. If it is a baby food thing, give it to the mum and again it needs to be checked and listed. If they don't have a special wrist band and are eating outside food then ask them to leave straight away as they have broken an important rule and risked the safety of others.
If you don't want it on their wrist you can get a table sign for them which you can attach branded paper with either the information printed on or written on. Could even do it on laminated stuff and use board pens.
It should be clear if people are bringing a cake in with a cake box or big bags. Also they may come in big groups or other traits that you can pick up on (if not start a spreadsheet to pick up on trends). Ask them directly if they look like birthday parties that if they wish to have cake it will need to be an inhouse cake or pay for cake age which can be collected as they leave. If caught in with a cake the whole party will be removed immediately.
Other option is that you have a pram and big bag area and lockers - if they try to bring in big bags, which they shouldn't need, that they need to lock them up. Have a staff member between food area and bag/pram area so they can stop them taking in food unless they have a preapproved wrist band. If they need a nappy bag, purse, etc then that is possible but this will stop them accessing food especially bigger items like cake.
Any one making a fuss point to terms of use of the venue and that if they wish to have a cake to blow the candles out, they will need to pay for that to be in a safe spot for fire alarms and to stop exposing those with severe allergies. They can also upgrade now to a "last minute party package" if available which will include x, y and z but in future that it is cheaper prearranged and would be x.
I've been to three soft play places with the separate pram/bag area and it's very successful at eliminating food or external food getting in. It might mean a reshuffle on the lay out but it might work wonders if you can do it.
When it comes to designing things for people you have four options cognitively: stick, carrot, undo or restrictions.
Stick is kicking them out of caught eating external food. You need to be strict with this and always stand by it no matter what "bad reviews" they might threaten you with. Go to the press with stories of "soft play centre trying to protect vulnerable customers attacked online by x" and you will find that you demonised those who are doing it for selfish means but want to protect those under your care.
Carrot is making it easier and cheaper for them to have your food or use your processes. Planned and Last minute party packages is this option.
Undo is not really an option here but it is to do with creating an environment where you can literally undo last action and correct it. You could do this in form of an official first warning that they can then go remove and clean up food, get an allowance wrist band (if relevant) or last minute package etc. Would need to enforce this however straight away literally taking them to the front desk/pay desk /party area.
Restrictions would be putting literal barriers to stop them bringing in bags of food from a separate area for bags and prams with bag checks for those ones going in (much like those at many London venues like tate, shows, etc). Been to a show recently where all bags checked for food and drink and as I had a medical issue I was allowed to bring in my bottle of unopened water. This is much like the "restricted access" windows that appear when you are on a computer etc. You can also spin these lockers and area as an added safety positive - no chance of someone stealing your purse or shopping if locked away! Or unsafe items damaging the soft play. One place asks if you have a pound for the lockers when you come in so it is clear that they must be used - this is a bouncy park so they have to be strict about shoes and valuables going into the bouncy areas, ball pits and soft play.
A mix of them together is you best option but might not be possible. Restrictions will make a huge difference and make it easier for you in the long run but might not be possible due to cost or layout.
I hope that helps. Without knowing a better breakdown of the layout I can't help much more.