Assuming that there is no minimum volume for a stable transition, and engineering is a one tonne minimum.
Automated would remove the need for a bridge.
So, maybe five tonnes.
As I recall, my manned design was twenty five tonnes as the smallest viable starship.
But current canon is hundred tonne volume minimum, and ten tonne jump drive default minimum.
LBB5'81:
Minimum drive size is 1Td from small craft design rules.
Computer is downrated 1 model number when replacing a small craft bridge; downrate it two for replacing a starship bridge.
This violates the following rules: the 100Td size limit, starship bridge size, and starship crew requirements.
Cylinder hull configuration chosen for space-efficient stowage. Consider using a streamlined hull configuration instead if space is not at a premium.
Code:
1Td MCr4 J6 Jump Drive (limited to J1 by computer size)
1Td MCr3 Pn 14 (1 EP) Power Plant (TL-15)
1Td MCr1 5G Maneuver Drive (Agility 0)
1Td Fuel
3Td MCr18 Computer Model/3 (effectively Model/1)
MCr0.7 Hull (cylinder, includes fuel scoops for gas giant skimming)
-------------Totals-----------------------------------------------
7Td MCr16.7
Performance is J1/5G, endurance is 4 weeks.
Assume sufficient autonomy to refuel at gas giants as needed to reach a programmed destination.
Consider house-ruling that the Jump Drive is hardened to operate on unrefined fuel, as it is a J-6 drive operating at J-1.
It could be up to 2Td larger (hull cost goes up by MCr0.2) and still have fuel for J1, but would then have only 4G acceleration. This 2Td additional payload could include fuel, each ton of which would provide an additional parsec of un-refueled range. (It's still limited to J1 by the computer, which can't be upgraded because of energy point requirements.)
EDIT TO ADD: This design elides fuel use a bit, assuming that the 1%*Pn power plant fuel use scales down. If not, computer is a Model/2bis and the fuel tankage is 2Td with no change to the cost. Fuel use changes to 1.25Td/Jump (Jump fuel plus 1 week powerplant fuel) with three weeks reserve fuel.