13 dwarves + the incoming picked reinforcements from Dain, whose expected arrival there is no reason to hide in a scenario where Thorin is open about surviving/being in possession of the Mountain, and already negotiating with Bard about what portion of the treasure should be considered the property of the survivors of Dale and their heirs.
Once Dain and his dwarves are inside the mountain along with the supplies they were carrying, the Mountain becomes that much more defensible, and can make ready for the next round of reinforcements. Dain's 500 were experienced warriors he was able to bring quickly for the express purpose of securing the mountain, not the total expected long term. That's part of the reason Bard and Thranduil were determined not to let him pass - if Dain and his people get inside, besieging or attacking the Mountain gets more difficult, even with superior numbers. The dwarves could and would open and fortify other gates and then bring in further reinforcements as the news spread. (Dain was the closest, being immediately to the east in the Iron Hills, with no mountain ranges or rivers between them and the Mountain. Thorin still has his people back in the Blue Mountains, who he presumably intended would join him once he had reclaimed their former home.)
I agree that Thranduil wanted the Mountain secure, but there's nothing in the text indicating he wanted to establish himself there. So I believe Thranduil would be unlikely to risk his own people to secure it if someone clearly not on the side of goblins or Sauron was already holding the Mountain, willing and even eager to bleed for it. The Mountain is nothing to Thranduil and his people but a strategic position, whereas the dwarves actually care about the idea of the Kingdom under the Mountain, which means they'll fight much harder for it.
If the risk you're referring to is 'risking his people in a fight against goblins when they show up', I don't see where Thranduil has much choice. It's nothing to do with keeping his word - once the goblins are on the scene in such force, it's imperative to defeat them, regardless of how much elves and dwarves may not like each other. The goblins are their mutual - and much more serious - enemy. Letting goblins get entrenched anywhere in the region is not in anyone's interest - elf, man, or dwarf.
I wondered if there might have been clashes between dwarves and elves toward the end of the Second Age or in the early to mid Third Age, but I can't find anything indicating that was the case. Everything I can find about dwarves' wars in the Third Age is them fighting dragons or orcs. The dwarves were on good terms with Eregion in the Second Age, fought against Sauron, and then retreated into Moria and took no further part. So maybe this is just one of those 'both sides have long memories and hold grudges like champs' deals?
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Once Dain and his dwarves are inside the mountain along with the supplies they were carrying, the Mountain becomes that much more defensible, and can make ready for the next round of reinforcements. Dain's 500 were experienced warriors he was able to bring quickly for the express purpose of securing the mountain, not the total expected long term. That's part of the reason Bard and Thranduil were determined not to let him pass - if Dain and his people get inside, besieging or attacking the Mountain gets more difficult, even with superior numbers. The dwarves could and would open and fortify other gates and then bring in further reinforcements as the news spread. (Dain was the closest, being immediately to the east in the Iron Hills, with no mountain ranges or rivers between them and the Mountain. Thorin still has his people back in the Blue Mountains, who he presumably intended would join him once he had reclaimed their former home.)
I agree that Thranduil wanted the Mountain secure, but there's nothing in the text indicating he wanted to establish himself there. So I believe Thranduil would be unlikely to risk his own people to secure it if someone clearly not on the side of goblins or Sauron was already holding the Mountain, willing and even eager to bleed for it. The Mountain is nothing to Thranduil and his people but a strategic position, whereas the dwarves actually care about the idea of the Kingdom under the Mountain, which means they'll fight much harder for it.
If the risk you're referring to is 'risking his people in a fight against goblins when they show up', I don't see where Thranduil has much choice. It's nothing to do with keeping his word - once the goblins are on the scene in such force, it's imperative to defeat them, regardless of how much elves and dwarves may not like each other. The goblins are their mutual - and much more serious - enemy. Letting goblins get entrenched anywhere in the region is not in anyone's interest - elf, man, or dwarf.
I wondered if there might have been clashes between dwarves and elves toward the end of the Second Age or in the early to mid Third Age, but I can't find anything indicating that was the case. Everything I can find about dwarves' wars in the Third Age is them fighting dragons or orcs. The dwarves were on good terms with Eregion in the Second Age, fought against Sauron, and then retreated into Moria and took no further part. So maybe this is just one of those 'both sides have long memories and hold grudges like champs' deals?