En H2, podría usar una columna calculada que tenga un índice único:
create table test(
biz_id int,
active int,
biz_id_active int as
(case active when 0 then null else biz_id end)
unique
);
--works
insert into test(biz_id, active) values(1, 0);
insert into test(biz_id, active) values(1, 0);
insert into test(biz_id, active) values(2, 1);
--fails
insert into test(biz_id, active) values(2, 1);