I don't know if there is an online service, but it's not that difficult to calculate.
If I take the values you show above, dimensions, snow zone, and roof angle, the load on each pillar on either side of the gate becomes 30.061 kN according to byggbeskrivningar.se
So we want a beam that can carry 61 kN of distributed load with a 4m span and with a maximum deflection of 13mm (span / 300 is the rule of thumb usually considered for acceptable deflection).
The formula for deflection of a freely supported steel beam with distributed load is 5 QL³ / 384 EI
Plugging our values into that formula and solving for I (moment of inertia), it shows that we need a beam with I of at least 1862 cm4
Choose, for example, an IPE200 (I=1943 cm4) or a HEA180 (I=2510 cm4).