In this case, fixed & variable costs are MECE for the reasons below.
Fixed costs are costs incurred by the business whether they produce 0units or 1million units. Yes, the more units they create, then logically fixed costs can be distributed among the units made but this is wrong thinking. We pay for salaries, rent, utilities: such fixed costs the business needs to incur before any unit is made.
Variable costs are costs that are DIRECTLY attributed to the making of the unit (material, labour,distribution etc) and therefore it is better to look at variable costs on a per unit basis.
If you do the chewing gum case, you will get 1 table full of costs. Indirect costs are therefore identified as your fixed costs.
You can look at both fixed/variable costs on a unit or total basis, but it all depends on the case, the business structure and what is being sold as this will dictate (per/unit or total) which methodology you use.
I hope this helps.