Does a lamp need a fuse?
No. It seems very unlikely there is any need for a fuse. The LIFX gadgets are presumed to be rated for direct connection to a mains branch circuit without any additional fusing.  The fuse was probably there to protect the switch as incandescent lamps do not require fusing. They act as their own fuse.

The "dimming" feature where it connected two bulbs in series is only valid for the original incandescent bulbs. It is virtually guaranteed to NOT work with any other kind of bulb, and especially not something like the LIFX.  You are quite right to remove this obsolete "feature".