The flow rate increases 100-fold (one hundred-fold) Would be a more idiomatic way of saying this, however, the questioner asks specifically about the original phrasing. The above Ngram search would suggest that a one hundred has always been less frequently used in written language and as such should probably be avoided. Your other suggestion of by one hundred times is definitely better than a ...
In a scientific or mathematical context they might occur, in which case you should use standard index form 1×10¹⁰⁰ for one followed by 100 zeros. There is the joke word "googol" for this.
As you note, "percent" means "for every hundred," so there is nothing at all wrong with percent values greater than 100 when discussing proportions, e.g. profits increased by 120%. Like fractions, however, percent values are longer to express than multiples, so for values much greater than 100 you may hear them less frequently:
The type of writing you are doing also plays into your decision. For example, in legally binding documents, like contracts or exhibits to contracts, the spelled out number is the legally binding number. So if a text said that, "you are 99% (one-hundred percent) responsible", the 100% number would be legally binding, not 99%.