Try using ReaStream https://www.reaper.fm/reaplugs/ set the identifier the same for the corresponding REAPER track, put a gain control after the ReaStream vst in BB so BB makes no sound only REAPER https://www.bluecataudio.com/Products/Product_GainSuite/ or LoudMax it's lighter than Bluecat but you may need 2 to cut the volume off, then chain up the FX in REAPER:

EDIT: if you need to run multiple ASIO https://www.vb-audio.com/Voicemeeter/banana.htm