It is possible to obtain this Achievement using Chapter Select... but only if you're willing to tweak the save file.
If you use Notepad (or similar) to open the save file at...
C: - Program Files (x86) - Steam - userdata - your id - 361550 - remote - data.xml
...you'll notice the following XML element:
(item name="ach_catman" type="int") some number (/item)
Change "some number" to "0", save the file, then
complete a level without falling in the water (I suggest Endless Sands). The Achievement should be waiting for you when you quit the game.
Note 0: Sorry about the file location & XML tag munging! ;)
Note 1: I don't know whether you have to have already completed the levels before you attempt this trick... I suspect that
may be necessary.
Note 2: The "ach_catman" element seems to keep track of the number of times you've fallen in the water on that entire "playthrough", which is why you normally have to start a new game to get the Achievement.