you might consider a thick inflated polystyrene box such as enzymes are delivered in with a large bottle of water at 37c or freezer blue plastic blocks equilibrated in a water bath to 37c should hold its temperature for more than 2 hours and the other lab could prepare bottles for you for the return journey
Yes you do need to maintain the pH but the cells will not grow much in 2 hours so just sealing the TC flask lid ( screwed tight and cover with clingfilm or parafilm) should keep the CO2 in the flask. Also when moving cells you need to almost fill the flask with culture medium to minimise fluid movement which will damage the cells if the contents move too violently. I think if you fill the flask with pre warmed and pre CO2 equilibrated medium at each end of the journey before setting out then the cells will be ok. Test this by leaving a flask of cells in the carrying box for 3 hours on the bench at work and check temperature and colour of the medium or even take a quick pH measurement