Is it realistic to hire a "taxi" or a marshrutka to go from Batumi to Akhaltsikhe and then to Borjomi via the mountain road?
The idea is to see the sights, those ancient bridges, etc.
How much may it cost?
Is it safe to travel this road in September?
When I was in Borjomi last month, I met a couple who had hired a taxi to go from Akhaltsikhe to Batumi via the mountain road. Unfortunately I cannot remember how much they paid, but it is certainly realistic to do so. They got the tourist office in Borjomi to find the taxi for them; you can probably do the same at the Batumi tourist office.
It is no more dangerous to travel the road in September than in high summer.
I drove this route in a Lada Niva.
It's not dangerous at all and not so mountainous as you might think. The biggest part of the road from Batumi to Akhaltsikhe is not asphalted but is wide enough to allow cars to drive in both directions without any risks and can be done with any kind of car / bus.
The part from Akhaltsikhe to Borjomi is in very good condition.
As a bonus I also advise you to visit the Green Monastery on the way to Borjomi.
See photos of this route on my page here: http://thdk.be/travel/georgia-armenia/rabat-fortress-in-akhaltsikhe-and-borjomi/
Apparently there's also a bus on that route: https://www.youtube.com/watch?v=16e_mJhmL0I
However I would not recommend local bus to tourists on this route. They are packed like sardines and then large luggage (e.g. mattresses) is squished in between and on top. Road is pretty bumpy too, so this option is really only for those who cannot afford anything else.
