I use B&Q Peat free compost, but you do need to riddle it first as it is full of large lumps of bark and the occasional flint. I am not impressed with any of the ones I have used to be honest, none of them are easy to use. Friend who grows huge numbers of bedding plants for a stately home darn sarf has just been forced to transfer to bark based and he is very unhappy about it.
Hundreds of ages ago when we used to go to Harlow Carr they were doing trials of peat free and peat based composts. I wonder what the results were.
The 'added John Innes' amuses me. John Innes is the name of a Horticultural Institute in California. They did a lot of experiments to find the best compost for various uses, seed sowing/cuttings..growing on..and long term. They came up with the John Innes Formulae which defined the amount of top soil, sand, peat and fertilizer to be mixed together for the said purpose. So what does 'with added John Innes' actually mean? Odd!
I suppose in the end as you are going to gravel mulch the bed the appearance does not matter, but it is annoying is it not?
Have you given any thought to the size and type of grit/gravel you intend finishing the bed with yet?