Can I use a 1:1 mixture of isopentyl alcohol and water as mobile phase to carry out paper chromatography of a mixture of red and blue ink?
The principle of chromatography is that you can always try (as per the question in the title). This can be beneficial: in my PhD thesis I had two diastereomers that were hard to separate on TLC using hexanes and ethyl acetate, but a hexanes/toluene mixture worked wonders.

However, in your case you should not expect better results. For one, as the commentators have pointed out the two liquids are not sufficiently soluble in each other. For two, your separation is already pretty good and it seems unlikely that it can be improved even more.