The stiffness of one's back is the ultimate arbiter of ability to pass under such a hurdle. The width of a low door is also vital if the the person navigating it can turn sideways to speed access and egress. Simply by managing a short sideways bow with the support of the stair handrails. Practice may help! A narrow door would force a longer, stooping, forwards shuffle. With the far greater risk of standing up while still under the door lintel.
There is really no need for the top of the hatch to be hindered by the present cross brace below the heavy top ring of timber reinforcement. [see below] A horizontal strut could just as easily reinforce the gap below the lintel.
I would not wish to enter the dome routinely by these doors but it will not be necessary. A normal sized, exterior door will be provided in the octagon wall down on the ground floor. Leading to safe, aluminium steps, with handrails, to reach the observatory, on the 1st floor, via a hinged, drop-down hatch. Only by closing the hatch can one access the veranda door. This is deliberate to avoid nose dives [no doubt with tuck, two half somersaults and twist] on returning to the observatory from the veranda. Nobody can hear your last scream in darkest, rural Denmark. Particularly in the middle of the night!
I tried laying 2x4, 2x6 and 2x8 timbers across the tops of the octagon's upright posts. 2x4" is too narrow to allow full miters to sit neatly on top of the 4x4 posts. Unfortunately I have little in the way of 2x6 left of any suitable length. Though I do have plenty of 2x8 looking for a useful purpose.
I could cut arcs on the outer edges of 2x8s. Which would be lag screwed to the tops of the octagon posts. Then an upstand of thin plywood, attached to these outer curves could help to seal the lower edges of the dome. Closing off the midriff gap where the support and dome rotation wheels will sit, otherwise fully exposed.
A skirt on the lower dome 'hem' is traditional to throw rain, dew and melt-water clear of the supporting walls. The upstand, rising from below, will aid this weather-proofing while still providing airflow ventilation to help avoid interior condensation.
Timber brackets, attached the the upright posts, would provide greater stability for the top ring of mitered 2x8 timbers. These same timbers would help to support the dome rotation wheels. As well as tying the tops of the octagon posts even more firmly together. All I need now is a stretch of dry weather to be able to work outside.
A useful suggestion on the CN forums was to clad the gores with plywood prior to cutting other ribs and gores away to form the observation slit. The plywood will stiffen the dome and avoid it sagging when structural material is removed.