Gene of interest number of copies in transformed K. pastoris
To predict the number integrations is probably not possible, but you can favor multiple integrations, if that's what you are after. At least in Saccharomyces cerevisiae integration of a first plasmid appears to stimulate the insertion of additional plasmids at the same site (1). It is therefore possible to favor multiple integrations (2) by increasing the DNA-to-cell ratio in the transformation, thereby increasing the chance that any competent cell receives more than one copy of the plasmid, which could then integrate in tandem. The easiest way is to simply use more DNA, but depending on your cell competence you might need to compensate this increase by using less cells for your transformation (or simply plate less cells on each plate) as you will probably also increase the overall colony yield.
One way to actively select for multiple integrations is to use a competitive inhibitor for the selection marker. Again, in Saccharomyces cerevisiae transformations with HIS3 selection can be combined with 3-AT (3). This allows you to inhibit HIS3 activity just enough to only permit the growth of cells that have integrated a desired number of copies of the plasmid. Of course, this method is far from fool-proof, as you might also select for mutants that have increased HIS3 expression for other reasons. Edit: I actually found one paper doing this in K. pastoris (4).
Finally, if your implementation allows it you can facilitate the screening for multiple integrations by including a fluorescent protein-expressing cassette in your plasmid. That way you can easily screen for different numbers of integrations by fluorescence microscopy or FACS.
Sources:

Orr-Weaver et al. (1983): https://mcb.asm.org/content/3/4/747
Plessis and Dujon (1993): https://doi.org/10.1016/0378-1119(93)90172-Y
Wikipedia (retrieved 15 Aug 2020): https://en.wikipedia.org/wiki/3-Amino-1,2,4-triazole
Menéndez et al. (2018): https://doi.org/10.2144/00295rr05