To get a private test will cost money.
The educational psychologist who covers your school should be able to test.
Learning Welsh will indeed be making things more difficult!
Keep in mind the possibility that he may
not be trying hard enough. And sadly, if he does prove dyslexic, he
will have to try harder than kids who are not. Although aware teachers can help, they can't do anything that will make it as easy for him as for kids who don't struggle with literacy however much the websites will tell you otherwise.
You yourself can find out if he prefers 'input' as visual, auditory or kinaesthetic. Many kids who struggle at school are kinaesthetic learners, who need to learn by doing. Of course humans evolved to do this, but classrooms are not good places for such folk and most teachers, who did well in the system by definition, are visual or auditory learners- and as you learn best, so you teach.
But if you know you can teach him ways to learn that suit him better.
I
would go back to the school and push for him to be seen by the Ed Psych.
Here is a kid-friendly VAK test:
http://brainboxx.co.uk/a3_aspects/pages/VAK_quest.htm