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The only type of ability that doesn't use the stack is a mana abillity, Mortician Beetle's ability is a tirggered ability and does use the stack. Paying costs for a spell, additional or otherwise, does not use the stack and is done up front as you cast the spell.

This is how your scenario would work:

You announce that you are casting Fling.
You pay all the costs, add mana to your pool if need be and sac Mortician Beetle as the additional cost.
Mortician Beetle's ability triggers and will be put on the stack on top of Fling.
Mortician Beetle's ability tries to resolve but can not find Mortician Beetle on the battlefield and "fizzles".
Fling tries to resolve, and uses the Last Known Information of the sacrificed creature as it was on the battlefield and deals 1 damage (pending any previously gained counters) to it's target.

Because Mortician Beetle is no longer on the battlefield when it's ability resolves, then it can't have a counter placed on it and Fling will 'see' it has having whatever power it had at the time it was sacrificed.

Thank you so much. I had looked into it a little further and found out that Mortician Beetle's ability does use the stack, just as you said. Originally, I had thought it did, but doubted myself. Once I learned it does, I thought of the same process you just explained. It's good to know I got it right haha. Once again, thank you.