Is there any way to become a Welder in the Army without being a 91E?

Think about the good aspects of leaning both jobs. It is all about gaining a skill. So by going the allied trade route, you will have more skills for a future resume.

91E is probably the best MOS in the motorpool. Mechanics have their TMs that guides them step by step, 91Es have their creativity and ability to think outside the box as their guide. During deployments you will more than likely be on a recovery team or working out of your welding trailer. Let me warn you that you will be asked to make trophies, grills, guideon stands and other dumb stuff way more than actual mission critical jobs. Some call 91Es the arts and crafts of the army I call them 91Everythings because they will do every job in the motorpool.

A you would be smart to learn both, as many welders and machinists do both in their civilian careers. That said your duties would be dependent on the unit assigned to, their work and the needs of the shop. Don't be surprised if you end doing light vehicle mechanic work or other duties within the shop. If at all possible during slow times learn the GCSS-Army system as that will be the future of the Army system. It will make you a better well rounded Logistician, when it comes time for promotions and or schools.