Screw Size for Bookcase

I would apply a high-strength thread-locking adhesive (e.g. loctite 270) to the inside of the head, reattach the head to the threaded rod (AKA stud), wait many hours, remove bolt.

LOCTITE 270 is a high-strength threadlocker for maximum efficacy in the securing and sealing of bolts, nuts and studs to prevent loosening due to vibration. The product serves to permanently lock assemblies which must not come loose. It works on all metals, including passive substrates such as stainless steel, aluminium and plated surfaces. It is proven to be tolerant of minor contamination due to industrial oils, e.g. engine oils, corrosion prevention oils and cutting fluids. Can be disassembled by heating to 300°C.

Other brands and products are available

From studying the assembly instructions, it seems to me it is also possible (perhaps even likely) that your furniture can be disassembled without fully extracting the threaded studs - so long as all the heads are removed.

^{With the head removed, the clearance hole allows the parts to separate}

After disassembly you can use the two-nut trick or locking pliers (vise-grip, mole-grip etc) on the centre of the stud to unscrew the studs.

I would suggest "brute force"; put in nuts and bolts. Although for a similar situation I have used 3 electric drills; One drill puts a pilot hole through both members, the second puts a clearance hole in one member only. the third drives a self tapping screw ( passing through the first member and threading into the second.)