Here's one which can handle 6: https://galactic.ink/sphere/#.

It's worth pointing out that if you have access to Adobe Illustrator it has a Kuler-type colour scheme building in the recolour toolset which will allow you to add as many pins to a scheme as you need.

EDIT:

OK - though I'm by no means a slavering Adobe Illustrator fan, for this task it does have a good toolset. See images below - less than five minutes to iterate a large number of variant colour schemes starting from your listed 14 colours base point.

Start by purging unused colour swatches - select the items currently coloured in your soon-to-be-base-scheme, and in the swatches palette, click the new folder icon -- this will populate a folder with all the selected item fill colours as swatches - name this something creative like Base Scheme.

Then select items to iterate colour scheme (in my case, the next row of squares) and click on colour guide...

And away we go - remember to make a new colour group each time before editing, and you should end up with a file whose swatches includes a set of folders with discrete colour schemes - and of course, this means you can later individually edit swatches as normal and the items with those swatches applied will update.

Hope this helps.