I would not move my bins.
In your place, I would have told my neighbour that I can no longer move the bins when they are full to the pavement edge for emptying and that they will need to stay where they are.
Could you put a screen of some kind round them? Perhaps use the kind of cheap wooden fencing sold in DIY stores.
If you or your grand-daughter cannot do this yourselves, nor know any handyman or woman who can, (has your grand-daughter a brother, boyfriend or father who could do the job?) go in and ask your neighbour if he will help with the job.
Realising that the cancer he hoped was in remission has come back is bound to be upsetting for him, and like most men, I bet he becomes grumpy when upset.
It is worth making an effort to get along with the neighbours, but shoving heavy bins about is not on, so see if you can find another solution.
If a screen round your bins isn't feasible, I would be tempted to plant a fast growing shrub or two in his line of vision from his kitchen to your bins - and if he complains about that, say nicely that you thought he would prefer to look at a flowering shrub than your bins.