After some advice please regarding whether easyJet will consider refunding the cost of rebooked flights (with them), following us missing our original flights the day before?

We were due to take a flight from LGW-KOS departing 14:50 however, it didn’t end up departing the terminal until 16:40 and take off until after 17:00, due to late arrival of aircraft into LGW from previous flight and ATC delays. We were travelling with two hold suitcases.

Entirely our fault we left our passports at home so got them brought to the airport by taxi. When we spoke to an easyJet supervisor at check in circa 2.5 hours before departure he said that as the flight was delayed to circa 1540 originally (when the aircraft was due to land at 1511 from previous sector), that if we came back to him by 1500 he would get our suitcases checked in and straight to security etc ie 40 mins before departure.
At 1545 when our passports arrived we went back to the check in area but a different member of staff was busy dealing with lots of enquiries from people and simply advised the gate was closed and we would have to rebook on the app, nothing else he could do before swiftly moving to the next customer.

We did have family members on the same original flight and so have a good idea of timing and explanations given to passengers throughout.

We therefore rebooked a flight the following day at cost of circa £550 for the four of us including children. We obviously then incurred overnight hotel accommodation etc given we live a few hours drive from LGW.

I am not after compensation as I appreciate it was our fault we didn’t have our passports originally but wondered whether easyJet may consider refunding the new ticket cost given we could have actually made the final departure time had their second member of staff assisted like the first one said he would as just to clarify we were ready at check in 55 minute before the final departure time from the terminal.

many thanks for any assistance.