I used to split my Lantus dose but it didn’t really make any significant difference, so I stopped doing it after 6 months or so.
Your problem sounds pretty much like my old problems with Lantus and I found the best cure was a pump – which isn’t much help perhaps 
I tried injecting Lantus earlier, later, larger doses, smaller doses but it was just too crude a mechanism for my wonky basal requirements. Said wonkiness is illustrated by my current pump basal settings:
00.00 – 1.950u/h
01.00 – 2.250u/h
05.00 – 1.750u/h
07.00 – 1.700u/h
10.00 – 0.700u/h
16.00 – 0.275u/h
19.00 – 0.500u/h
22.30 – 0.725u/h
I need more than double the background during the night than I do during the afternoon – this was just impossible to get right with Lantus.