But I know that at the end of the transaction chain the character in our little orange drama who will be paying the tax is the hapless consumer.

But that's incorrect. Payment of the tax will be allocated among the market participants in accordance with their price sensitivities. It's almost certain that no single party will pay all of the tax.

For example, suppose the consumer is extremely price sensitive (econ-speak for saying they change their behavior a lot in response to price changes). They have lots of readily available alternatives to carrots, all of which they like just about the same as carrots, and which aren't much more expensive than carrots are now. Meanwhile, assume the farmer is extremely price insensitive - he's got a lot of sunk costs in land that is really maximized for carrot farming, and carrot-specific improvements and machinery, etc. In that scenario, the farmer would end up paying most of the tax. The price can't be increased very much on the consumer (else they'd stop buying carrots and move to alternatives), and the farmer doesn't have a lot of options. So the farmer ends up eating the extra cost, selling his formerly $1 carrots for maybe $0.91 or $0.92, and the consumer ends up paying only $1.03 or so for the carrots.

If the price sensitivities are reversed, the outcome is reversed - the more price sensitive farmer would bear less of the tax, while the price insensitive consumer would eat more of it.

It's simply not true that either the hapless consumer or the producer or the merchant (or the importer or shipper or wholesaler or anyone else in the transaction chain) will end up paying all of the tax.