Cita: "Jarcek"No there is not a typo. Franz II ruled until 1804, when he became Franz I. Fact that Austria/(an mint) lived a dream a while longer or it was economically inefficient in war years does not change that fact.
Do you have a reference that the rule of Franz II ended in 1804?
The history I was taught was that he appointed himself Emperor Franz I of Austria in 1804 but his reign as HRE Franz II didn't end until 1806.
I'm traveling so I can't cite references from my paper books, but here are some online references:
https://rowman.com/ISBN/9780761870777/Double-Emperor-The-Life-and-Times-of-Francis-of-Austria
https://www.britannica.com/biography/Francis-II-Holy-Roman-emperor
https://en.wikipedia.org/wiki/Francis_II%2C_Holy_Roman_Emperor
https://en.wikipedia.org/wiki/Holy_Roman_Empire
https://en.wikipedia.org/wiki/Dissolution_of_the_Holy_Roman_Empire
One paper reference I can cite: SCWC, 1801-1900, 8th edition, page 76. (And previous editions.)
And then there's the coins themselves. Look at the legends of the coins for the types issued from 1804 to 1806. Here's the 1 Thaler:
https://en.numista.com/catalogue/pieces33724.html
The pictured coin is from 1806 and it the legend says is it issued by Franz II, Roman and Austrian Emperor. The earlier types say HRE Franz II, the later types say Franz I, but the1804 types say 1) he is still Franz II, and 2) he hold two emperor positions.
You say that the coin lettering is a "dream", but it could be argued that all positions of HRE were a dream. This particular dream was shared by everyone in Europe and beyond. The Wikipedia article on the Dissolution of the Holy Roman Empire shows a newspaper clipping from August 1806 describing the then-current event.
See the references in the Dissolution, there are multiple 1806 citations to published books.