Yes, the first sentence is correct:

Mon fils travaille comme promeneur de chiens depuis quelques semaines. -- My son has been working as a dog walker for several weeks.

The present tense is used with depuis in many cases where in English you'd use perfect continuous tense. There's a lesson on this:

Using the present tense (Le Présent) - and not the compound past (Le Passé Composé) - in sentences with "depuis" (since/for) in French (French Prepositions of Time)

The construction a été travaillant is a literal translation from English and doesn't work in French at all. As a rule, try not to translate word by word and, instead, train yourself to think in French.