The value of the loan is equal to the sum of the discounted values of the repayments.

∴ b = ((1 + r) (m + (1 + r)^n (r s - m)))/r
and m = (r ((1 + r)^(1 + n) s - b))/((1 + r) ((1 + r)^n - 1))
where
s = present value of loan
m = periodic repayment
r = periodic rate
b = balloon payment
n = number of periods (or payments) before the balloon
There is no formula for the periodic rate r
. You will need to solve for it using one of the above equations.
s = 377100
m = 5998
b = 206540
n = 35
Solving s = (m - m (1 + r)^-n)/r + b/(1 + r)^(n + 1)
for r
∴ r = 0.00372937
So the effective annual rate is (1 + r)^12 - 1 = 4.56819 %
This assumes full amortisation, i.e. 35 payments of 5998 and a final one of 206540 at the end of month 36.
"With full amortization, the amortization schedule has been set so that the last periodical payment comprises the final portion of principal still due."
Addendum
With 36 payments of 5998 and a payment of 206540 at the end of month 36
s = 377100
m = 5998
b = 206540 + m = 212538
n = 35
∴ r = 0.00425237
The effective annual rate is (1 + r)^12 - 1 = 5.2239 %
Confiming with Excel
