Interesting question,
A SSR is not necessary! Why is the first PIC constantly reading the EEPROM? Will it impact upon it if the EEPROM cannot be read? i.e. whilst the other PIC is reading it?
Your link is for a USB switch, again, probably not suitable.
I presume you are using the SCL and SDA lines to read the EEPROM? well, one way to do this would be to have 2 sets of 2 transistors maybe as a simple switch, where the 2nd pic controls both sets, so switches the first PIC off and switches itself on, and vice versa.
I guess since it is voltage levels this would work?
Wilksey