Products that I am trying to weld are HDPE and LDPE plastic with approximately 125 degree centigrade melting point. Space available for welding is very restricted because I have a bag and a cap and there is no way to access inside the bag to apply pressure from inside and outside. We used ultrasonic welding machine but because the alignment between the bag and cap has little change for each bag there are some bags that are not quite fit together and weld break under pressure. So I searched for a suitable replacement to ultrasonic. Now I think I need a procedure with cost efficiency and without so much tolerance to worker’s low accuracy.
This picture is ultimately what I’m trying to achieve but I have no idea how.

One option is laser welding, which is very clean and fast but it is expensive and again prone to misalignment. Another option is heat and which is, I think, my option. But I have no idea how to do heat welding this clean and without so much waste of material.
Any suggestions?