Solenoid Simulated Recoil
The amp-turn product is what counts for the solenoid. 
ie number of turns x amps drawn from battery.
For a given voltage, if you double the turns using the same wire size you also double the resistance, so the current halves and the amp-turn product remains the same :-(.  For a given voltage, to increase the amp-turn product you must increase wire size, and or wind the same total turns in multiple coils in parallel. More on this below.  

6 x AA batteries in series (=6S) will give about 9V unloaded when new. You need Alkalines if you want the batteries to provide any difference compared to the cap. 

AA alkaline batteries may provide 4A - 6A on near short circuit when new. 

At 0.3 Ohms the required V for 6A is V = IR = 6A x 0.3 Ohm = 1.8V - so the batteries have more than enough voltage but current capacity is lacking. The low coild resistance will load the batteries down to around 2V while dissipating most of the energy in the battery's internal resistance. 

AT ~= 4V fully charged a nominally 3.7V Lipo will try to deliver
I = V/R = 4/0.3m~= 13A.
That's about 2C for a 6Ah LiPo so it probably manages the 13A OK.

Placing the 6 x AA all in parallel = 6P which could give 4A x 6 =  24A short circuit BUT lead and contact resistance will reduce that. And the 0.3 ohm coild resistance will only draw 4 or 5A at 1.5V :-(.

Arranging the 6 x AA as 2S3P  = 3 x 4A = 12A at 2 to 3V loaded -> which is about the capability of the LiPo. At 3V the coil will draw about 10A when the battery is very new and only for a short period. 

About now, investigating a capacitor looks like a good idea. 



Energy in a capacitor = 0.5 x C x V^2 
Charged to 6V a 1000 uF capacitor stores
Energy = 0.5 x 1000 x 10^-6 x 6^2 = 0.018 Joule.
That's 0.018 Watt seconds, or at say 3V = 0.006 A.s
Or say 6A for 1 ms, or 1A for 6 ms or ...
Change that to 10,000 uF and energy is 10 x larger. 
100,000 uF is beginning to get useful - 1.8 Watt seconds or about say
1A.s at 2V. 

A 2V+ rated "supercap" of 0.1F or higher with suitably high discharge rating charged by 1 or more AA in parallel may start to be useful - as long as its specification allows a suitably high current draw.

At 1.5V I = V/R = 1.5/0.3 = 5A initially = too low.
Higher V gives higher I but you'd need a suitably voltage rated supercap. 



Supercaps around 1F MAY have high ESR and low peak current rating.
A search through selection tables will locate ones that are suitable.

As an example only, this Vishay (reputable brand) 25F 3.0V capacitor from Digikey at $US 4.11/1 in stock, datasheet here is rated at 25A at 65C.  

At 0.3 Ohms at even 3V you'll get 10A - so well under current spec for the capacitor and "nearish" to the LiPo value. You can use two x 3V caps in series, with suitable balancing of voltage, to get higher current.

At 10A initial a 25F cap will drop from 3V to say 2.8V in about 0.5 seconds - so for pulse use, if 10A is good enough, this cap may do what you want.

More horsepower!:

If you want more "kick", the same wire wound in two parallel winding will give you 0.075 Ohms and about 40A at 3V -  too high!
A little playing with wire size and turns will allow a configuration with more amp-turns and still inside the capacitor's specification and double the recoil. 

Example only:

Imagine that you have a coil with 100 turns which give 0.3 Ohms resistance.
If you apply 3 Volts then current is I = V/R = 3/0.3 = 10A.
So amp-turns in solenoid = N x I = 100 x 10 = 1000 amp-turns.  

Now using the same wire wind two coils of only 50 turns each connected in parallel.   Each coil now has a resistance of 0.15 Ohms (0.3/2).
The current in one coil is now I = V/R = 3/0.15 = 20A.
The amp turns from one coil = N x I = 50 x 20 = 1000 At = same as before.
BUT you have two coils in parallel providing 1000 At each so together you have 2000 At = double what you had before with the same wire. The current is 4 times higher so for producing A.t this is "less efficient", but, it's double with the same wire.   

The current in this case is above the supercap's maximum current rating.
By adjusting turns and wire size you can get varying results. 

Below At = A.t = ampere turns = amps in coil x turns in coil. 

If coil area and cost allow you can get an  increase in At by a factor of N (approximately) with the same current and voltage by simultaneously increasing both turns and wire cross sectional area by a factor of N. 
Resistance increases by N with increase in turns by a factor of N.
Resistance decreases by N with increase in area by a factor of N.
So resistance is the same so current is the same.
But turns have increased by N so amp-turns has increased by N.
I say approximately as increase in wire area will alter how turns are packed relative to core so effective At may not vary by exactly N in practice.