The answer to that question doesn't change anything. The assumption is that the security of one's salvation being divorced from one's acts provides no barrier to self-destruction or even self-indulgence. "If I do 'x', I'm still saved" is the general statement that describes the position. The answer has to be "yes" if one's salvation is not resting upon one's own works. The answer is "no" if one's salvation IS resting upon one's works. That simple. So the alternative to security of salvation apart from works is works salvation.
The one who presupposes works salvation will argue that the one who has this security apart from their actions is then free to do what they want. While true in one sense, it ignores the fact that the one who believes in salvation apart from works and is born again is not the same person they once were. They are changed and it is necessarily the Spirit of the Lord prompting their thoughts and actions. That doesn't mean they will be perfect, but it does mean they won't be looking for any excuse to indulge themselves.
So the fact that the answer to your question is "Yes. Based only on the fact that one is not saved by works nor lost after being saved by works, suicide would not take away their salvation" does not negate the fact that the real point of salvation makes this a straw man. At least in the instance where someone really is born again.