I understand that the perfect solution to this problem is a trust, but
I am not sure I am at the point in my life where I can set one up.
Setting up a trust would probably cost on the order of $1,500-$2,500. As you acknowledge, a trust is probably the best solution to your problem.
I'd also recommend thinking a bit more about the big picture.
Ultimately, if you are first to die, your partner if going to have to meet all of the household expenses, meet all of the needs of your child, and pay off all of the student loans that income based repayment does not forgive in 20 years. And, without you to contribute to the household's finances, your partner is probably going to have to make choices that will increase your partner's income and as a result increase the amount of the income based repayment of the loan which will reduce the benefit provided by that option.
Making your plan more complicated may save your partner $10,000-$30,000 of principal payments over twenty years in a best case scenario, but possibly much less, but at the cost of making it very complicated to deal with.
It may be more straightforward to simply leave all of the life insurance to your partner, eliminating the need for trusts or distinctions between your child's needs and your partner's needs that could give rise to legal liability, and eliminating the stresses involved with paying student loans and being a single parent at the same time.
You may also want to think about getting additional life insurance beyond $250,000, as that seems pretty meager to cushion your partner from the burden of having to raise your shared child alone for 14 years and beyond into higher education, without your financial support. Paying $100 a year in more premiums would allow for a bigger death benefit that would solve the problem you are trying to address in a beneficiary designation by fully paying off your partner's student loans, for less the cost as a trust, with money left over.