Of the following methods, using a tube with 4 untabbed cells compressed within it, with cooling holes if needed, would probably be the cheapest and easiest solution. Strong foam or springs at each end only - preload then latch in place, should work well. "Latch" can be as simple as a disk pushed into place then a pin pushed across outside it. Or a stiff wie (say paper-clip thickness+ which can be pushed through cross holes then bent.
Springs are a bad idea unless you can guarantee a holding force which exceeds the spring force under all situations.
4 in a tube: If you place 4 batteries in series in a tube and then apply some force at both ends you should be able to make a battery that can withstand any sensible applied force and which will fit inside your baton.
If you find a plastic or even perhaps cardboard tube just slightly greater than the battery diameter you can if necessary cut holes in the tube sides at numerous locations to allow cooling. This should not be necessary at modest discharge rates.
Standard "nipple ended" untabbed cells should be able to be used this way. Batteries can have a spring or stiff foam wad at each end. Batteries are pressed into tube with spring etc at one end then an end disk added then a pin or wire or screw across the whole tube to keep batteries in place and maintain spring compression.
Diagrammatic illustration of scheme below.
Contacts not shown - add end disks with wires.
No inter-battery connections required (friction contact).
Foam or spring pressure must resist acceleration on impact when dropped longitudinally.
Tube can be as strong as desired
Black - body tube.
Orange - foam or springs to provide compression.
Red - disk, fitting body diameter. Non conductive probably.
Blue - pins or wire that push through holes in opposite sides of tube.
Insert foam + disk + pin at one end.
Insert contact (not shown).
Insert 4 batteries.
Insert contact + 2nd foam + 2nd disk.
Push down until 2nd pin can be inserted
Bend pins over for security OR use screw + nut - say 2mm to 3mm.
Nylock nuts for extra points.
Tube needs to be strong enough to withstand foam compression forces + dropping + ...
Tube MAY need to be ventilated. Probably not.
Tabbed cells?: Using AA batteries with tabs will allow you to solder a pack in the shape of your choice. Tabbed cells are usually (not invariably) rechargeable. From the bluetooth module datasheet it seems that 4 x NimH would be too low. BUT if you can bypass the BT onboard regulator you MAY be able to run on 4 x NimH.
Lithium Ion: Energy content in 4 x 1000 mAh cells (see below) is about 4 x 1000 mAh x 1.2V (say) =~ 5 Watt.hour. A single LiIon 18650 cell delivers about 2000 mAh x 3.6V = 7+ Watt hours. IF you can use the voltage that a LiIon delivers then a single 18650 rechargeable would do well.
Premade: You can buy premade NimH packs of 3 and 4 cells made for cordless telephones.
Your bluetooth module needs 5V at a current which varies widely with mode.
Bluetooth datasheet here.
Your report says that the cheapest batteries lasted 50 hours.
At 2000 mAh, 50 hours = 40 mA average.
At 1000 mAh (more likely for cheapest batteries) = 20 mA average.
4 Alkaline or Zinc batteries will NOT provide 5V when flat (fully discharged). 5V/4 = 1.25V = 50%-60% discharged.
The Bluetooth module has a linear regulator. Vout is not specified BUT if it is 3V3 then 3 batteries could be used with an LDO down to about 1.133 V cell. (3 x 1.1333 = 3.4V = 3.3+ 0.1V headroom worst case. (Very tight).
@David Kessner 's advice re potting PCB etc sounds good and in earlier years I would have agreed that that sounded the best way to do it. My experiences of the last few years indicate that you can get superb results more easily.
If you need better than superb, try potting.
We have found that a semi floating PCB works wonders.
We have a light with the PCB along the long axis and mounted in slots with an end cap, ability to move slightly fore and aft and some modest padding at ends. It has a few SMD ICs and a small ferrite core inductor on it plus discretes and some glue components. You can drop th ABS cased light 100 x from 1.5metres onto concrete all axis, or say 20 x from 2 to 3 metres or say 5 x from 5 metres without damage, in many trials. The 5 metre test reliably breaks part of the housing but nothing else - fall energy must be enough to reliably impact it past breaking point. Light still functions but you can no longer hang it up ;-).