I still think it's hasty and presumptuous to decide that the problem is either the TK70, or the batteries, without further data.
The problem could be either, and further tests would be required to narrow it down.
What if you exchange the TK70 for a new one, and it turns out you had a couple of problem cells causing the trouble all along?
You are unlikely to get sustained Turbo using any alkalines, they can't maintain the high current.
Good NiMh cells fully charged
should be able to maintain the current, but if one or two of them are no good, the voltage of the whole
battery would drop quickly, causing exactly the effect described in the OP.
When the load is removed, the open circuit voltage of the bad cells could bounce back, so even a voltage test after removing the cells from the flashlight might not necessarily identify the bad cell/s, if there are any.
To the OP: Do you have any friends with a hobby charger? Because here is a way to tell if the cells are good.
Charge them on your 808 the way you usually do,
Then one by one discharge them at a high rate, like 5A on the hobby charger and chart the discharge curve.
Have a look at the voltage that the cells maintain under discharge.
That will show you if they are keeping sufficient voltage under load (>1.2V) or not.
If all the cells perform well under that test, then you'll know you have a faulty TK70 and can then return it.
If you don't have any friends with a hobby charger, and you don't have a 6V power supply, then unless you can think of another way to track down the cause of the problem; you can't just assume it's caused by the flashlight or the cells.
Here's an example of some D cells discharged with a 4A load I did quite a while ago.
The Voltage levels shown are not exactly right though, because there was a significant voltage drop through the wires and connection to the cells.
You need to take a measurement at the cell under load and compare it to the hobby charger's voltage reading and the difference is the voltage drop, which should be added to the discharge curve voltages. (Edit: added chart with corrected data based on V drop and % of error in mAh readings that I have since calculated.)