Just a thought....this most important part of the tiny tremelo, can be fairly easily replicated with a bog standard linear POT.....to help establish if any of the the clicking comes from shunting the pickup's current signal to ground quickly, or if it's 555 related.
Take a 1M pot, short one of the outer lugs to the centre lug & the the other free lug to ground (you've now got a pseudo LDR...a rheostat) - connect your guitar output to the pot (ie hot to to the two lugs shorted together, guitar ground to your gound lug)...feed onwards to your amp.
Have a play, moving the pot wiper around, fast, slow, pulsey-ish (hehe, I've made a word up) etc. It'd be interesting to hear how the tone changes as the pot gets towards the ground end of its travel (bad I'd imagine!)....and this method allows slo-mo.
i noticed the other day that the ticking goes when i turned the guitar volume down,(try it) so i figured something on the input might help!!!!..and it did!.
That kinda ties in with my line of thought that I touched upon a couple of weeks ago…..
If you roll the volume pot off a little on the guitar then that provides the necessary in series resistance for the LDR/rheostat to work as a potential divider.
Edit: When you back off the volume knob on your guitar, you're also putting extra resistance
in series into your circuit (guitar pickup->Vol pot top lug-> pot's resistance track->Vol pot wiper lug-> Tiny tremolo circuit input)....whereas with your latest mod, you've just placed resistance in parallel with your circuit's input. There will be parallel resistance ...the ratio of series resistance to paralell resistance will depend on the wiper position of your guitar's volume pot. For example a 250k volume pot, half way backed off would be 125k in series with the LDR & 125k in parallel with your circuit's LDR. (assuming a linear pot, which I realise most guitar volume pots are log!)