If you feel you can cope with the housetraining and walking etc - then I would say "get the pup". Microchipping is about £15 - the breeder should have done this before the pup leaves its mother, but of course s/he may not be vaccinated, which will be a bout £50-£60.
I would suggest that you get a dog if you can, rather than a bitch, then if you are unable to afford neutering , or can't face the thought of your pup going under an anaesthetic, then it isn't so serious. Yu can leave him entire (though you still have a responsibility to keep him away from other people's bitches. There is actually quite a lot of evidence that neutering isn't as(medically) necessary for males as for females.
You don't say what breed the pup is - only that it's a lapdog. Will it need professional clipping/grooming? This can be expensive and cost about £50 every 6 weeks to 3 months, depending on the breed.
If you decide to go ahead I hope that you find that the pup stimulates your interest in life again, and that walking and training it helps you to feel better.
I don't blame you for finding your children wearying rather than stimulating - I did too when I had severe depression. I loved them to bits but I didn't have the energy to cope with their natural liveliness and naughtiness - children are hard work! But so are young puppies - please don't forget that. Sometimes housetraining seems to take forever, and even when you think you've cracked it, there can be a relapse.
I have to admit that we got a pup last year, and I'd forgotten how much hard work they are - we never stopped for a moment - cleaning up, feeding, training, more cleaning up, getting her to stop eating furniture, cleaning up again - it has only just started to calm down and she is over a year old now.
But best of luck whatever you decide.