People are probably more familiar with the phrase "lipstick on a pig" to convey a futile attempt to change the true nature of something. But, I like the imagery of of a bulldog better.
There's an unnecessary negative connotation associated with calling anything a pig. It's not critical that the underlying thing that someone is trying to change be ugly. It's more about it being difficult to make any changes in the first place. Using a bulldog instead makes it clear that you'd have to put in a lot of effort to wrestle it to the ground to even be able to apply the lipstick. And ridiculousness of having lipstick on a pig or on a bulldog is effectively the same.
All in all, the bulldog does a better job of expressing that both the process and result were misguided.