Why 4‐methylidenespiro[2.4]heptane cannot be converted to 1‐chlorobicyclo[3.2.0]heptane?
Here the concept of angular strain comes into the picture. The ideal tetrahedral and is around 109° 29'. As the angle between the C−C bond increases, it is more away from the ideal $\mathrm{sp^3}$ hybridization and hence will become more unstable.

For a cyclohexane molecule, the angle between the C−C bonds is 120°, therefore the deviation from the ideal angle is 10° 31', whereas for a cyclopentane the angle between C−C bonds is 108°, therefore the deviation from the ideal tetrahedral angle is around 1.5°. This is a very low deviation.

Since the transition is from cyclopentane which has a relatively lower deviation from the ideal angle to cyclohexane which has a relatively higher deviation from the ideal angle, the above transition does not take place.

Plus the number of carbons are not balanced in your reaction.