With Elon Musk’s commercial space company poised on Monday to announce recovery plans from last month’s dramatic rocket launch failure, its more-established rival is battling economic and political challenges that are in many ways tougher to overcome.

Skyrocketing launch costs over the years combined with controversy over longstanding dependence on Russian engines have forced United Launch Alliance LLC, the U.S. military’s premier rocket supplier, into a strategic reassessment that already has shuffled its leadership and altered...