you will probably need to arrange the chainrings further apart than normal, in order that you can run cross chained without the chain scraping on the side of the 44T ring. If this distance is large enough it will likely cause troubles with indexed front shifting. To deter the chain from jamming, you will likely need to arrange ramps that will guide the chain from the big ring to the small chainring.
Because of its small size, a 12T chainring will sit over a typical BB axle but not over the BB shell, or EBB cups. This means that the chainline of the 12T chainring will have to be ~38mm or more, and with the increased spacing the big ring will probably end up with a chainline ~48mm or more. These minimum values assume that you are using some kind of RH BB bearing that sits flush with the end of a 68mm BB shell. If the BB uses typical modern (eg HT-II) cups that protrude another ~12mm, the chainlines get pushed out that amount too.
The other things to worry about are
1) rear mech capacity; (you will use 32T of it in the chainrings alone, which does not leave much for the cassette) and
2) wear; even a very hard 12T chainring will wear quite rapidly in use; there are just not enough teeth and maximum tooth loads will be incredibly high.
On the latter point if you bring your bodyweight to bear on a typical length crank with a 12T chainring fitted, you can cause chain tensions of around 800kg to be generated. This is only a bit less than the breaking strength of many modern chains, and any persistent use of this kind at any significant angle will often cause the chain riveting to become unreliable.
Needless to say there are several potential problem areas with the arrangement you propose, which may be challenging to address; there are reasons why this is not done already....
cheers
