First, check with both carriers to assure that the aircraft will take your specific seat in J. While the short-haul is likely 3-3 seating with the middle blocked, not all lie flats can accommodate a car seat and the child will need to be buckled into their seat on departure, arrival & enroute turbulence.

Second, If you can get by without the stroller for the trip, consider purchasing on arrival or ordering & having delivered there. If that does not work, my vote is for a cheap umbrella stroller and a bag (both available on Amazon). The bag is not strictly required, but will keep the stroller from getting filthy.

That then leaves you time to find a permanent solution on arrival. Shipping is an option, but unless a third-party is paying, my guess is that you will do better locally.