Its probably not what you want to hear, not least because we all love buying bikes...but you can do lejog on any bike.
Just pace yourself accordingly. The roads are not challenging, the hills not that demanding and the weather, well it will be British.
I did mine with discover adventure, who organised absolutely everything. I spent about a tenner in the entire two weeks, it was all included.
If it helps to prepare...or motivate, check out a few blogs...here's mine
https://www.cyclechat.net/threads/lands-end-to-john-ogroats.155855/