Parallel battery supply
First of all, PlasmaHH in the comments gives a good idea on how to do this better...... However if you REALLY want to do it this way (to answer this particular question)

First of all, use schottky diodes (less Vf and at these voltages, that drop can be significant especially with the input voltage on these boost regulators). Second, may want to consider using something different. Looking at the specs (which it doesn't give much info on) it says INPUT current is 1.2A. Please correct me if I am wrong people, but that is not the same as output current!

The thing it does say about output current is: "The maximum achievable output current is approximately proportional to the ratio of the input voltage to the output voltage." And it sounds like you are wanting 1A output current from this, which I don't think is achievable by looking at this graph:


If you look at the BLUE line (1V input even though yours will be less because of the diode drop) the efficiency is awful, even at less than 200mA. This wil probably explain why it struggled to light an LED.....

Anyway, the short answer is, either rethink your input or use a different IC to boost the voltage. 

(feel free to critique my answer if I have missed something or got something wrong as I am still learning electronics myself and am always happy to learn! I can edit the answer if I have accidently given any false information)