How can I change my buffer system for protein purification?
Assuming that your protein can handle being in the new buffer, you have multiple options.

Dialysis will work although you might want to consider using a centrifuge spin filter with a suitable MW cutoff to gradually switch buffer and concentrate the sample. Be aware that some of your protein will stick to the filter, giving a loss of sample.

Alternatively you can perform size exclusion chromatography to switch into the new buffer, but you better be sure that the sample will remain stable or you may clog the column. If you can perform affinity chromatography, you can also opt to change buffer in this step.

You can of course try to start with a Tris based buffer and simply adjust pH by dialysis in a tris buffer with the desired pH. Just make sure that the dialysis volume is large enough to counteract the buffer in the sample.

Instead of using PBS you can also try with TBS (Tris-buffered saline), see wiki for details. https://en.wikipedia.org/wiki/Tris-buffered_saline