MusicBee does have support for ASIO (see screenshot). It should be selectable in Prefs/Player/Output. Be sure you are running the latest version of MB.
If it says you don't have ASIO,
re-register the ASIO drivers in Windows. I had to do this once with MusicBee, either a MB patch or a Windows Update made it not "see' the drivers. The code below, if copied into a text file, given the *.reg" extension, and ran will re-register Realtek ASIO drivers but you'll have to find out what your own are named (look in Device Manager) and replace the code accordingly:
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\ASIO\Realtek ASIO]
"CLSID"="{A80362FF-CE76-4DD9-874A-704C57BF0D6A}"
"Description"="Realtek ASIO"
[HKEY_LOCAL_MACHINE\SOFTWARE\Classes\CLSID\{A80362FF-CE76-4DD9-874A-704C57BF0D6A}\InprocServer32]
@="c:\\windows\\system32\\rthdasio64.dll"
"ThreadingModel"="Apartment"
[HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\ASIO\Realtek ASIO]
"CLSID"="{A80362FF-CE76-4DD9-874A-704C57BF0D6A}"
"Description"="Realtek ASIO"
[HKEY_LOCAL_MACHINE\SOFTWARE\Classes\WOW6432Node\CLSID\{A80362FF-CE76-4DD9-874A-704C57BF0D6A}\InprocServer32]
@="c:\\windows\\SysWow64\\rthdasio.dll"
"ThreadingModel"="Apartment"
