How do you share and document structured on the job training/skills assessments?

Hi Miranda! Thanks for posting your question here! You got me thinking! Have you considered creating your assessment one question at a time? Phil Mayor created this self-assessment example which I think is similar to what you're looking to do. I hope it gives you some new ideas!

You might try https://h5p.org/cornell-notes or the documentation tool https://h5p.org/documentation-tool and modify them. These are open-source assets that if you join the community you can get help to change the code to do the calculations you need. For example multiple selection question has code for + or - points  Then you give them back for everyone to use.

You wrote "mobile friendly...where the trainer can pull up a list of specific/measurable tasks and check off proficiency, resulting in a score for the trainee in LMS". If you are looking for a trainer to do the skill assessment and enter it into the LMS, then the LMS admin can simply set up a course called digital skills, assign the instructor to it, who can create topics and levels and input the records directly (provided the LMS allows this on a mobile device).   

I like the self-assessment example mostly because it enforces self-ownership of learning.  What I would recommend is that you write them like learning objectives (with standards) to ascertain competency level (beginner, intermediate, etc). ie. "In Microsoft Word I can format and adjust cell padding in tables, can set tabs to align bullets or paragraphs anyway I desire, and can correct accessibility issues identified.".