Can we start a sentence like this : "As well as that, ........." (meaning "In addition to that/Besides, ......")? Or "Apart from that,............" is better?
The meanings are different:
'As well as that'" has, as you quote, the meaning 'in addition to that'
and is used:
"Let's have another look at that First Prize! And here's the key ...to your very own new Mercedes!...courtesy of Dean's Autos, the car specialists. As well as that, we want you to have - yes - one year's free petrol!! How about that folks!!"
" True - he can be a bit aggressive when he's got a few drinks in him, but apart from that, he's a really nice guy."
In the first, there is an accumulaltion - this as well as that.
In the second, we are asked to set something aside, disregard it, not to take it into consideration.