'Table-tennis racket [Amer.]/racquet [Brit.]' seems to be the formal technical term, at least according to the International Table Tennis Federation:
> Regulations > 2.04 The Racket
2.04.01 The racket may be of any size, shape or weight but the blade shall be flat and rigid. http://www.ittf.com/ittf_handbook/hb.asp?s_nu...
In conversational use, though, I agree that that may not be very idiomatic, since rackets normally have strings, as in tennis or badminton.
In fact, 'table tennis' isn't the normal term either, 'ping-pong' is, at least in AE, except among purists. So I'd suggest adding 'ping-pong paddle [Amer.] [ifml.]' as well. But not *'table-tennis paddle,' which would be a hodgepodge of diction levels.
'Bat' in this context is probably unknown to most AE speakers. For us, bats are found only in baseball, cricket, and belfries. :-) However, judging from a search at the USA Table Tennis Federation (http://www.usatt.org/index.shtml
), 'bat' does seem to be used occasionally in competitive table-tennis circles, especially in the phrase 'hard bat.' So it might be safer to mark it [formal] or something than [Brit.].