In the last couple of weeks (by utilizing someone else's code that was somewhat better at this than mine) I tried again and this time I completed the search. So I can now state that the four largest 3-balanced factorization integers are:
1149570826484083326976 = 2^55 * 31907
1208735110046668895744 = 2^9 * 53 * 239^7
2205036970648967839744 = 2^18 * 1531^5
4546782683595318279169 = 7^10 * 2003^4
The largest 4-balanced factorization integer is:
19260075803546226131439208984375 = 5^18 * 7 * 947^6
Which creates an interesting situation. We now have the largest and smallest k-balanced solutions for k = 1, 2, and 3. (My brute-force find of the smallest 3-balanced factorization integer is here.) And while we also have the largest solution for k = 4, we'll likely never know the smallest!