You will need to extract the data into an array of objects:
B4X:
Sub BitSetToArray (set As B4XBitSet) As Object()
Dim o(set.Size) As Object
For i = 0 To set.Size - 1
o(i) = set.Get(i)
Next
Return o
End Sub
Sub ArrayToBitSet(o() As Object) As B4XBitSet
Dim set As B4XBitSet
set.Initialize(o.Length)
For i = 0 To set.Size - 1
set.Set(i, o(i))
Next
End Sub
This array can be serialized with B4XSerializator (RandomAccessFile library).