I'd recommend using an angle grinder to make the crack slightly wider, then gun in some Toupret Fibacryl. It is a caulk that is reinforced with fibres. Given that it is water based, it will shrink back slightly. After a day or two, then fill with Toupret Touprelith F.
Materials will cost about £20 excluding sandpaper and paints. You will have a lot of materials left over though. Hopefully you will be able to use them elsewhere.