Here is some info on dissolving times of some brands. Unless you dump very soon after using the toilet it looks like any brand will be okay since the worst ones dissolve in less than a couple of minutes.
For 11 full timing years we used whatever TP my wife liked and had no problems.
Brand...........................................Dissolving Time in Seconds
(lowest is best)
Charmin Plus 1-Ply..................................9
MD 2-Ply...................................................9
Coronet 2-Ply..........................................12
Angel Soft 2-Ply......................................12
Scott 1-Ply..............................................17
Soft 'N Gentle 1-Ply ...............................18
Soft Weve 1-Ply.....................................20
Marcal 1-Ply...........................................20
Soft 'N Gentle 2-Ply...............................21
Seventh Generation 2-Ply.....................21
Quilted Northern Super 2-Ply ...............69
Charmin Ultra 2-Ply ..............................77
Cottonelle 1-Ply ....................................96
Charmin 1-Ply .....................................180
The TPs with an Excellent disintegration rating from a Aug 2000 Consumer Report article are:
Albertson's Soft Choice Ultra
Safeway Select Softly Ultra
Quilted Northern Ultra Soft
Quilted Nothern Soft Prints
Target
Kroger
Angel Soft
Charmin Plus with Unscented Natural Aloe
SoftWeve