I'm pretty sure there aren't any rules for this in the Core book. What would you do if a player wanted to use two one handed weapons? Perhaps two short swords, or two small axes, or a sword and a dagger. I was thinking excess Tengwars could grant a bonus attack with the secondary weapon? Just an idea.
The question with that is always, how elaborate to you want or need to go?
the easiest way to incorporate them would be to treat the off hand as a shield in terms of mechanics adding a parry bonus (using load as the the scaling factor as with shields).
A more offensive variant would be to add the load value of the off hand to the main weapon injury rating (similar to 1h vs 2h use of a long sword)
Both variants lean on existing mechanics for weapons in combat, which basically removes the need to think about balancing.
You could technically use both by selecting one or the other by the currently chosen stance in a combat round.
The idea about extra attacks is interesting, but it may require thinking a bit about combat balance. While in cooperative games balance between players isn't as important as in competitive games, you still should think about where your math is heading.
Not sure about the specific probabilities, but for this case looking at something of a summary load limit for both weapons might be a good starting point - something in the order of 3 or 4.
Aside from math, with the limited combat options of the system there is also the question if giving a single char regularly more attacks than the other players will be seen as balanced purely in player activity terms. And if the second attack doesn't happen often for the player ( because of luck or the chosen trigger requirements) then his 2 weapon style might feel underwhelming, If that is an issue depends more on the players than the house rule itself and how they approach combat and the game itself.
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Also, (maybe this should be another topic), why is it so much more expensive to train using skill and adventure points (during the fellowship phase) than it is when you create your character? I think the cost changes, right? p118ff of the core book show the costs they are a lot more than when you start out.
I think because they simply wanted to create different effects for the prior experience without the need to tie them into the experience scaling, in particular they wanted to scale combat vs generic skill a bit differently.