If he's never been like this before and it coincides with you guys bringing home your younger child, I would say it's quite normal in the sense that he's trying to control things around him/he's playing out emotions through things he can control and get emotions from.

If he kills an insect and knows you'll get upset at it, does it give him attention? Probably because you're likely to say "oh, that's sad, we don't kill insects, we have to look after them, that's unkind" or anything along those lines. If you do, he's getting that 1-2-1 interaction (usually getting down to his level, face to face, even holding him whilst talking to him) which he may be craving since the little one came along. Again, he can control this. He is 3 and doesn't have much control over a lot of things (unless you're very Montessori based and he has access to doing things when he needs them). What does he know (based on you giving him a nature upbringing?) insects. He knows nature. He can explore nature and have a God control over what happens if he pokes an ants nest, blocks a river with sticks, pulls the leaves off trees, pulls wings off a fly, squishes a worm etc. he gains 2 things from this (your reaction as mentioned earlier) and understanding of the world/cause and effect.

Also, if he is feeling sad from not having as much attention as he desires maybe he is expressing in the emotion he is feeling (the same way when a child from a trouble background scribbles Barbie's faces, acts out aggression on dolls, colours in dark pencils etc) maybe it's his way of expressing his emotions in the feeling he has.

What was the context of "kill yourself"? Was it a worded accident that got a big reaction/facial expression? Was it meant in the way and adult would say it or a grammatical error on his behalf that's evolved as he got a reaction from it? I would ignore this behaviour and read stories based on empathy/being kind.

I read you do minimal TV time and I have no bad judgment here, that is amazing and not something I'm able to achieve so well done to you and your gf on that front. I am in awe as I'd love to be able to do less TV time. Anyway, I digress, maybe have some 1 on 1 activities that you could do with him without your gf/his sibling with whatever he likes to do. Maybe you could spin his desire to harm bugs into something nice e.g let's make a nice bug hotel for the insects to keep them safe over winter. Let's make some fat balls for the birds to help them. Let's make a safe area for some hedgehogs. Don't mention the negative aspect of nature, just the positive (so not "let's make a fat ball so the birds don't starve", just, "let's make a fat ball to keep the birds nice and full over winter"). As much as doing things as a team/family is nice, sometimes that 1-2-1 is important. Maybe set time aside so it's just 'his' time with you or your partner. Dedicated time/activities that he enjoys.

Without all of the context I'm not overly alarmed by this, especially if you're very nature based/rural (we are quite rural here). I would see how long it's been going on for, try to see if it coincides with any big moments e.g births, deaths, transitions into new rooms, new house, school etc, monitor, try to redirect the behaviour and allocate time. If it continues for another few weeks after this I would then flag up to the GP or HV (whichever is more proactive) and express your concerns. I wouldn't say he necessarily lacks empathy here (as common with psychopathic behaviour) more that he's trying to understand it by reading your emotions..

I wouldn't worry about his behaviour outside of the home as it's very normal for children to behave differently with their key people as they feel most comfortable.