Should not be a problem as long as the cells contact the heat spreader with no more insulation thickness than the existing cells' sleeving. The batteries run and charge cool until the very completion of charge when the cells start to get quite warm. The power manager is looking for this temp rise over ambient as a signal (one of several, the manager is also monitoring voltage and keeping some running estimate of charge state from a time and current drawn/delivered standpoint) for switching from high rate charge to trickle charge. I think the heat spreader is not for heat sinking, but rather sampling the temperature of all of the cells and bringing this sample over to the strip PC board where a temp sensor is seeing the temperature of the air near two adjacent cells plus the temperature conducted to it from the heat spreader to the PC board to the sensor leads. Whatever cells you use should come with insulating shrink sleeving and can be taped down onto the heat spreader. Then, to help hold down the cells, small foam strips can be put on top to fill the gap to the top cover. Or, there's even enough room to put an inverted homemade springy battery socket thingy that incorporates the original safety overcurrent/overtemp devices and makes contacts to those economical retail store tabless cells. There has been forum discussion elsewhere on the merits of not soldering directly to tabless cells, and some have even built spotwelders using big caps like those used in monster car audio stiffening applications. Leave the vents of the cells and case unblocked in the event gases need to be vented.
One unknown is whether the smaller AA cells will have sufficiently low internal resistance to handle the PB1400 worst case current draw which I am guessing is hard drive seeking plus spinning up a CDROM. Wouldn't be good to get a low battery sleep from this. I have read somewhere that the very highest capacity AA cells get this by trading off internal resistance. Some experimentation, or even better browsing in the RC forums for test results, would be good. The remote control and robot wars people push their cells very hard indeed on both charge and discharge, and I have seen, somewhere on the net, results of comparison testing of cell vendors. In fact they push the cells so hard that many of them just solder direct, figuring that any seal compromise is irrelevant since they plan on wearing out the cell's plate material and insulator integrity before they dry out from seal leakage.