Ensuring SQL Server Trigger Execution

It can be done, using the following template:

CREATE TABLE tablename ( ... );

/* for direct invocation */
CREATE FUNCTION propagate_data(newrow tablename) RETURNS void
LANGUAGE plpgsql
AS $$
BEGIN
    INSERT INTO other_table VALUES (newrow.a, newrow.b, ...);
END:
$$;

/* trigger function wrapper */
CREATE FUNCTION propagate_data_trg() RETURNS trigger
LANGUAGE plpgsql
AS $$
BEGIN
    PERFORM propagate_data(NEW);
END;
$$;

CREATE TRIGGER propagate_data AFTER INSERT ON tablename FOR EACH ROW
    EXECUTE PROCEDURE propagate_data_trg();

In an ideal world you would have two choices, SNAPSHOT and READ COMMITTED SNAPSHOT (RCSI). Make sure you understand the basics of transaction isolation levels before you decide which is appropriate for your workload. Specifically be aware of the different results you may see as a result of moving to RCSI.

This sounds like it's not an ideal world as you don't have any control over the application that is generating the select statements. In that case, your only option is to enable RCSI for the database in question such that the selects will automatically use RCSI instead of READ COMMITTED.