I have a TV1J on a MM WO and it is perfect with a 2xAA config, but dimmer with a 1xAA. I don't think that 1xAA can keep up with the power demands, especially when your Vin is 1.5v regulated up to 3v. Again, if you manage the Vf of the diode (the perfect Luxeon for 1xAA is the R2H/R3H as mentioned above). The TX1K is being under driven, with its higher Vf and the current constraints of a 1xAA config.
As long as you drive the 3W at 1W specs, it will ideally perform very much like a 1W (keeping in mind that most Luxeon III's will have a slightly higher Vf). So if you want a 1xAA light, it would be better to stick with the R2H/R3H 1W Luxeons (I have not yet come across an H-binned Luxeon III).