The highest I got thus far, after migrating my FD to a few BR in Penang and Ipoh, is 4.16% for 12 months.    Please advise which BR branch gives higher FD rates?

No, not necessary to open a saving account.
Unless if you want the monthly earned interest to be paid into your saving account,
then you will need to open a saving account. But this requires a minimum of RM 20K FD deposit.

If use board rates, overall yes. BR FD rates are the best.
Except 5 years FD, MBSB is slightly better than BR.

But MBSB 5 years FD rate is straightly not negotiable, regardless of the FD amount.
Meanwhile for BR 5 years FD, the rate is negotiable if the FD amount is big enough.

You have to keep in mind both Bank Rakyat and MBSB have no PIDM protection.

But sometimes, banks with PIDM have better FD promo rates than Bank Rakyat board rates,
so we have to keep an eye on recent banks' FD promos.

If you can wait, or your RHB FD maturity date is after this Wednesday, then just wait
until this Wednesday to see what is OCBC new FD promo.

BR's 12 months FD rate is negotiable if the amount is fair enough for BR. So you have to try
to ask for a better rate, otherwise BR will give you 4.01%.

Usually BR will give you gift(s) even if you do not ask for it, but still it is better to ask for it.