Is the lowest rate always less expensive?

No.
The reason is that each type of card has its own fee. The low fee only
applies to a certain type of card. Most processing companies sell “low
fee” programs. These low fee programs are deceptive in that there are
additional fees that the merchant has to pay for rewards, business and
corporate cards. These additional fees are never explained to the
merchant.
Example. Your low fee is 1.39%. If you process $10,000.00 in sales and
your low fee applied to everything than your total fee cost would be
$139.00 which is $10,000 x 1.39%.
At the end of the month your fees were $441.12 which comes out to 4.41%.

What happened to the low fee of 1.39%. These low fee programs are “loss
leaders”. The processing companies will lead with this low fee. It is
attractive and sold as the lowest fee possible. These low fee programs
have surcharges for cards that do not qualify for this “low fee”. The
surcharges can be 2% or more on top the low fee you were quoted. Now you
are at much higher cost and are paying much more than you should.