Does a piece that counts as two token types (e.g., both a Badlands and a Beasts via "Settle Into Hunting-Grounds") have a double impact on effects scale with both token types (e.g., "Beset by Many Troubles")?

Does a piece that counts as two token types (e.g., both a Badlands and a Beasts via "Settle Into Hunting-Grounds") have a double impact on effects scale with both token types (e.g., "Beset by Many Troubles")?

No. Each piece either counts for the effect or doesn't count for the effect; there's no way for a piece to count multiple times.
It's akin to counting up the number of rocks that are white or red – a rock that's both white and red would only be counted once.