With a 3" diameter water heater draft hood you can vent in up to a maximum 7" diameter lined chimney (there are some other important variables required to perform this chimney flue size calculation
). Your plumber has already confirmed that it would be too large with that 3" liner removed and in order to keep using that chimney, it needs to be re-lined (chimneys cannot be used without liner anyway).
You can do it yourself for as long as your local building department will allow you to. They might simply say that the installer has to be licensed to perform such work.
Assuming that they let you do it, there are two things that could make self-installation difficult: the shape of your chimney (some of the older chimneys run partially on an angle – corbelled), and access to the top opening – tall, unsupported chimney would most likely require scaffolding installation – you have to get there from the top, and you should not lean the ladder against it, it’s dangerous.
I would suggest to insulate the liner if possible (your jurisdiction might require it), check your local hardware stores for prices, size the liner accordingly to its length, number of bends / there should be tables provided with the WH and / or liner manufacturer. Watch this video - chimney flue
The other options…
Unfortunately they are more expensive because you have already purchased the water heater. If it can be returned, and exchanged for a power vent equipped water heater or a tankless type:
The new WH with installation would most likely be over $1000.00
The tankless type WH with installation close to $2000.00 (or more depending on the model).
Natural draft type water heaters cannot be converted into the power vent operated appliances that discharge exhaust gasses horizontally / through the side wall – the appliance must be listed for such type of ventilation, and your local building department wouldn’t allow it.
Let me know if there’s anything else I could help you with.