Peltier thermal electric generator power output
I am new to the Peltier thermal electric generator scene and I am wondering if you could recover some of the electrical energy needed for the cooling of a refrigeration system by using another Tec as part of the heat dissipation by connecting the second Tec to a battery charger?


Paradoxically, yes, it could be done, and without violating any laws of thermodynamics.
However, the efficiencies are liable to be so poor as make it not worthwhile in any but the very most extreme cases. In such cases there are superior means that TECs and TEGs to achieve this task. 

TEC  = Thermoelectric cooler - electric energy in - heat and coolth* out.
TEG = Thermoelectric generator - heat and coolth* in, electric energy out.

*Both devices have a temperature differential between two surfaces. You can think of heat and coolth being used or generated OR of heat energy flowing between the two surfaces.

TEG: The MAXIMUM efficiency that a TEG can achieve is the Kelvin efficiency = (Thot-Tcold)/Thot with temperatures in degrees Kelvin (or Rankine). Low cost TECs when used as TEGs (*Peltier effect devices ) usually have a maximum allowed hot plate temperature not much over 100C. Devices made as TEGs MAY employ materials and construction methods allowing temperatures of up to about 200 C.  Even at Th=200C = 473 K and Tcold =0C = 273 K the Kelvin efficiency is only (473-273)/473 = 42%. In practice it is usual for actual efficiencies to be a small to very small fraction of Kelvin efficiency.   For low delta-T (temperature differential) even the Kelvin efficiency is low eg Th 50C/323K, Tc = 10C/283K. Z_Kelvin = (323-283)/323 = 40/323 = 0.124 or about 12.5%. The achievable TEG efficiency would be unlikely to be more than say 6%.

TEC: TEC COP (ratio of heat energy pumped to electrical energy in) is liable to be in the 0.25 - 0.75 (E_thermal = E_electrical x 25%-75% ) range depending on quality, temperature differential,  energy transfer rate and more. Generally COP falls as temperature differential rises and is maximum at a low but not minimum intermediate energy transfer rate. 

An example will demonstrate what the above leads to.

Take a 100W input TEC producing 50W of coolth (COP = 0.5) with Tin = -10C and Th = 30C in an ambient of 10C. Th>>Tambient due to heatsink efficincies.
Cooling power = 50 Watts.
Electrical input = 100 Watts.
Heat output = 100+50 = 150 Watts. 

Take a TEG operating from Th = 30 C/303 K to Tc = 10 C / 283 K (excellent heatsink).  

Energy in to TEC = 100W, energy out of TEC = 150
Energy into TEG = 150 Watts
TEG Kelvin efficiency = (303-283)/303 = 20/303 = 6.6%
Actual TEG efficincy is maybe 3.3%
TEG electrical output is  3.3% x 150 W ~= 5 Watts  

Energy saved by TEG = 5/100 = 5%
In a situation with a very high delta-Temperature TEG efficiency would rise substantially and in niche cases you might get a situation where the TEG dT is greater than the TEC dT and even better results could be achieved. But even with a TEG operating between say 200 C and 10 C Z_Kelvin = 33%, Zpractical is maybe 15%-20% and electrical gain is perhaps 20%+. This is seldom liable to be worthwhile.