Starting at 7th level, they can learn the Reconstruction Mutagen:
Prerequisite: 7th level
For 1 hour, at the start of each of your turns, you regain hit points equal to your proficiency bonus if you have at least 1 hit point, but no more than half of your hit points.
Part of the description of the Mutagencraft feature reads:
As a bonus action you can consume a single mutagen, and the effects and side effects last until you finish a short or long rest, unless otherwise specified.
Due to the 1-hour duration of Reconstruction, the effect would outlast the end of the short rest, effectively pushing the number of active mutagens to 3.
Much later, at level 18, this becomes much more interesting due to the Exalted Mutation feature:
As a bonus action, you can choose one mutagen currently affecting you to flush from your system and end, then immediately have a mutagen you know the formula for take effect in its place.
This enables you to switch the 1-hour duration mutagen to a long-term one, at this point effectively enabling you to be under the effect of up to 5 long-term mutagens at a time.
The description of a short rest specifies the activities one might take during that rest:
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
Consuming a mutagen in my mind would count as drinking or eating something.
As an extra bit of info, the Mutagen would lose its potency only after the end of the short rest, according to the Mutagencraft feature description:
They are also unstable by nature, losing their potency over time and becoming inert if not used before you finish your next short or long rest.
Am I missing something here? Is it possible for an Order of the Mutant blood hunter to consume a Mutagen towards the end of a short rest?