Think about one more factor: your first review will be in Dec/Jan. If you join later you will have less time to deliver something meaningful on your level (and expectations from lvl6 are very high). On the other hand if you join very late (e.g. in August) and join the team after Oct 1 then your first review will be automatically MA, so you will have ~9 months to deliver something meaningful (before your real first summer review).
If that’s the case, by starting in June, OP has to do one extra month of work and only gets paid base salary + bonus for it. If RSUs are e.g. 50% of income, that’s like working an extra month and only getting paid 50% for it.