Is there any way to push the bundle products (created under Shopify Bundle) to Shopee? While managing the inventory?

The app doesn't natively support syncing a bundle directly to Shopee.

Best way around bundles is to just push the individual products from Shopify into Shopee, and create a Shopee bundle instead. The order sync will still work since the bundle contains each individual line item of the product in that bundle, which will sync up to your inventory.