If you have a good feel for the AMM range, and you have a speed advantage, you can try dipping in and out of the max range of the missile - the enemy will fire off some missiles, then you run out of range and watch them evaporate.  There are a few problems
 

• (1) knowing when they have fired the AMMs (you need a fairly big res 1 sensor for this)
  
• (2) making sure that you are fast enough to get back out of range before the AMMs reach you - this is made even more difficult if you have an ECM advantage as this will reduce the range that the enemy can fire the AMMs but does NOT reduce the AMMs endurance

I have had some sucess with this against PD bases around planets, but I've also lost several fleets when i've made mistakes on the approach.  Ive never suceeded with this against a mobile fleet (I rarely have sufficient speed advantage)

I also just realized that armored missiles might be a decent option, depending on your salvo size.

Thanks for the input, guys!

I had recently started on cloaking tech, so that was something I was considering but didn't know what to expect from cloaking. . .  based on some positive feedback that it might help in this kind of situation, I'll push that research a little harder.

Also, since I can sit so comfortably at 20mkm (what happened to 'Gm' by the way? I guess it's too unfamiliar and strange, hehe) and farther I was thinking I really do need some missile tech to lob back at them.  The result though will be a bit unsatisfactory in that it will just drain their AMM.  Though if they have more of that than the missiles I bring to play, or if they have some energy-based PD I could still be in for a rumble.

So. . .  Plan A: I'll try like Panopticon says, or similiar to blue emu's #1.  Though I'll try to make missiles which are 'real', as much as that will be possible within a few years starting from zilch for missle tech. . . !

Plan B: The Klingon path.

BTW, these guys are my neighbors in Ross.  The other three jumps from Sol are barren, and I'm having Duranium issues.  I don't like the idea of hostile neighbors who can obliterate me.  Not my comfort zone. 

If you are fighting things that outrange you and that dont miss, the kill limit is your only option.   Any close assault fleet is going to end up in a situation where it has to grin and bear it, and hope you have enough numbers and toughness.   

Alternatively you could just wait for them to hop through a jump point you are sitting near. .  or shut down your drives and hope to ambush them.  ^^

@Blue Emu - that just sounds like massively out teched. .  Oo.  Not much you can do there, ever.   

There is a way to do this without exploiting, simply stay out of their range, AMM are generally short ranged and even relatively low tech missiles will outrange them. Use small salvos that they have to reply to and when they start running off to reload then close the distance and blow them away with your lasers.

I admit that would take a bit of time if you don't have any missile tech at all, but needs must as they say.

Alternatively, you could throw wave after wave of your own men at them until they reach their preset kill limit and shut down, then kill them with whatever is left.

Just my opinion, but I find out'thinking' the AI in terms of fire control assignments to be an exploit. Even if they'd have only one fire control on their entire ship, they wouldn't actually stick all fifty AMM tubes onto it for a single missile.
On the other hand, using AMMs as offensive use is a last-resort kind of thing I find myself doing every now and then, too. I've also read it in various fictions and it's strategically and rule wise perfectly fine and entirely un-exploity.

In your position I'd go with anything of the following: A yet even larger PD fleet, armoured decoy ships, engagement range fleet movements to slowly wear them down with minimal lsoses, too many ships to care about the losses, or perhaps try a stealth and / or minefield approach.
From a roleplaying perspective I'd avoid a fighter suicide run, but mentalities differ.

If you haven't done so, turn on SM mode and check the space-time bubble box on the contacts tab of the main window. It'll cut down a lot on your waiting time (the game will only advance time in that system rather than the whole galaxy, so a lot less work for the game to do).

As for dealing with AMM swarms, it's a tough one. I've had some success using heavy shields because they regenerate as you advance, but it's hard to fit enough of them. Lots of point defense spread out between your fleet and set to fire at point blank, and if you're using lasers for your main battery you can try setting them to area denial mode if they have decent range.

There are a few ways to deal with this sort of thing.

1) Soak up their fire with decoy missiles.

Design a slow, super-long-range missile, and build a ship with active sensors and fire controls to match. Sit way back... over 100 m-km away... and start firing one-missile salvos at the enemy. You might even want to put active sensors on your decoy missiles, to make sure that he sees them. He should start firing AMMs at them as they come close, and since you are only firing one missile in each salvo, he will waste 53 of his own AMMs to kill each decoy. With a combat-load of a few hundred of these decoys, you should be able to run him entirely out of missiles.

Of course, this requires the tech to produce sensors and fire controls with a range in excess of 100 m-km.

2) Soak up their fire with Fighters.

A bit suicidal, but you can build hundreds of cheap, small, fast fighters and pack them into Carriers, equipped with one ASM each. Launch them from well beyond your opponent's range. Each Fighter should draw a full salvo (53 missiles) before it dies. If he fails to notice them and kill them, then you can send in hundreds of single-missile ASM salvos, which should be enough to kill him since each enemy FC can only target one incoming salvo at a time.

5) Soak up their fire with armored Civilian ships.

Use your civilian shipyards to build a few ships with nothing but engines, twenty-four (or so) layers of armor, and a one-hull-square active sensor (one HS is the largest civilian sensor allowed). Send them in ten or twenty million km ahead of your fleet. They should soak up thousands of rounds of AMM fire. A bit exploitive... but so is an avalanche of AMMs.