In this case, I would pursue the damages through the LID first. If they deny fault, you can submit it to your insurance company. You can also inform your insurance company of the situation, but you do not want to file a claim if you do not have to. A claim will increase your rates for up to five years which is why it's better to let the other company payout for the damages. If they claim to not be at fault your insurance company can deal with that for you.
It should be noted that your umbrella coverage only steps in whenever other insurance limits have been reached, so it isn't likely that this would be the case in this situation.
For more information on filing claims and umbrella policies, check out the link below:
Good luck and if you have any further questions, don't hesitate to ask