Is it permissible to continue conversing with a non-Muslim young man online with the intention of inviting him to Islam, while adhering to Sharia controls and not overstepping boundaries in speech?

Our advice to you is to avoid speaking with males unless it is a necessity, as it is a gateway to fitna (strife/temptation) and evil. Satan may beautify what is forbidden today, so it is necessary to block the means (to sin). You are not responsible for this person's anxiety. If you wish to advise him, let it be by guiding him to websites that discuss Islam and its virtues, so perhaps Allah may guide him.