OK - very good.
Let me summarize where I think we are at, as there are really a couple of possibilities;
1) This could very well be the MAIN BASE UNIT - that is starting to fail - and causes the loss of communication between the Main Base and the cordless handsets.
2) These eBay handsets are also an "unknown factor", in that they will NOT register to the main base unit. BUT (see #3 below)
3) This is an "older generation" system, that was not really designed to be "easily expanded" to more than the number of handsets that come in the package with the original system.
I mention numbers 2 & 3 above because these eBay handsets MAY be fine, BUT they were "left over" or "salvaged" (used) handsets from a KX-TG5433 system - because you mentioned that each of these handsets was labeled as "#3".
Also, the current generation of Panasonic system IS EASILY expanded (up to 6 handsets), and the current handsets even have the capability to "de-register" THEMSELVES - so that they can be registered to a different Main Base Unit of the same (or compatible) model.
As an example, let me give you a link to the search results for a "Panasonic cordless phone 3 handsets" I did on Amazon, and have you look at what is available - and the current street prices for such a system. Most of these results can also be looked and change the number of handsets too, to sometimes get an even better price.
Panasonic cordless phone 3 handsets - Search Results (Amazon)
Click here--> http://amzn.to/1FBBvi8
Once you take a look at some of those system, you may want to COMPARE what the cost of the eBay handsets are, plus batteries, etc., and then see if it might be about the same cost (or maybe less) to replace your system with a new one.
Also, if the eBay handsets can be returned for a refund, assuming you have a reasonable seller that will take them back, then those funds could be put toward a newer system.
I am NOT trying to pressure you into a newer system, but just suggesting you may want to "take a look" at what's out there - as another possibility.
Please let me know what you find or if you have any more questions.
Thanks, ***** *****