There are three things this could be.

1. Mortar joints between the flue tiles are missing. If a camera was ran up the flue and mortar joints were in tact still then this is not the problem. Wood smoke is acidic it reacts with the lime in the mortar and the sand falls out. When the mason lays the clay tiles he just used the same mortar used to lay the brick. Usually a type n or s mortar. It is supposed to be a high temp refractory cement but a lot of masons don't use it. If the mortar joints are missing between the flue tiles there is a smoke crossover between the gaps within the chimney. Causing the issue. This normally happens in reverse though. Basements are low pressure zones in the house air likes to flow to the negative. and a chimney is a good place for make up air within the home. This is the scary part. If the mortar joints are missing and the smoke is crossing over through one it's probably happening with the gas unit too. But instead of the form of smoke it's carbon monoxide.

2. Smoke chamber. Same theory is applied here as well. Wood smoke is acidic and smoke is passing through the gaps between the bricks because the sand has fallen out of the mortar

3. Smoke crossover at the top of the chimney. This is most likely not the problem since it normally happens in reverse. (make up air comes in the basement) easy way to test this theory though. Turn off the pilot and block off the top of the flue and see if it smokes.

Smoke chamber is probably the issue and 800$ isn't a bad price. There is typically a lot of labor involved and 2 people have to work on it. One inside and one outside. Unless the sweep has a chamber parging system.

Aaron's Chimney Service, thank you! This was a great lesson and maybe a lifesaver for this household; the last two sentences of #1. I will immediately call upon another chimney specialist and thoroughly review these three items with them. Again, thank you for the time!