Transitions needed for dividing two fixed integers
Using a Turing machine compiler I wrote I devised a program to do this in $O(c\log(q))$ transitions assuming $0 \le p &lt; q$. More precisely it does it in $2 \lfloor \log_2(q) \rfloor +\lfloor \log_2(p) \rfloor + C$ states for some value $C$. I'm sure the value I found(~$409$) wasn't optimal.

The actual program itself is too long to include here(totaling $3584+18\lfloor\log_2(q)\rfloor + 9\lfloor \log_2(p) \rfloor$ lines) Here is the program in my pseudo language for approximating 5/9, it can be compiled with the program found here. The syntax for the language is also found there.

xn=1,4
while i decr {
    xnH=xn
    a=xnH
    a=a,0
    t=0,1
    while xnH decr {
        p=9,4
        while p decr {
            incr(a)
        }
        incr(t)
    }
    two=a
    two=two,2
    two=two,0
    first(two)
    while a decr {
        decr(two)
    }
    xnH=xn
    b=two
    b=b,0
    while xnH pop {
        b=b,1
        xn=xn,1
        incr(t)
    }
    pop(b)
    xn=xn,1
    while xn decr {
        p=two
        while p decr {
            incr(b)
        }
        incr(t)
    }
    xn=b
    incr(t)
}
xnH=xn
xnH=xnH,0
while xn decr {
    p=5,3
    while p decr {
        incr(xnH)
    }
    incr(p)
}
incr(xn)


xnH holds the final value(the whole thing is assumed to be after the decimal point). It shouldn't be too hard to extract out a single digit from it(in $O(1)$ states). Also as a note even for small values of i this program takes quite a long time to execute.