I took my LNAT this year, but I'm yet to receive my results, but I got into the two LNAT schools I applied to. These are just some few tips for you to consider. What worked for me may not work for you, but you can always give it a try.
I didn't do any tutoring or enrol in any LNAT course. It isn't even advisable on the website that you don't do this because it doesn't really help. It's like IELTS; there's no way to study for it, you can only take practice tests.
I used only one book called "Passing the National Admissions Test for Law" by Rosalie Hutton and two others. I can't say it helped, because all I used it for were the practice tests. Regarding the Critical Thinking books you mentioned, I took Critical Thinking during my foundation year and the first two books were on our reading list. They didn't even help with my Critical Thinking module, not to talk of the LNAT. Use it if you wish, but I wouldn't recommend it. I also used Prepgenie LNAT past questions but they were quite outdated, because they had 14 passages each, instead of the normal 12.
For extra tips, I would say check the deadline for LNAT among the schools you wish to apply to, then fix a date when you'll be emotionally and mentally stable to write it, then start preparing towards that date. Don't feel pressured to take it too early or too late and try not to panic on the test day, or the day before, although I did. If it helps, form a study group so you can all encourage and educate each other.
You can start practising without setting a time limit, just to see how long you spend and how well you can think, then you can go on to take the test under the real exam conditions. Try reading the passage before the question, and then vice versa, to see which one helps you understand better and work faster.
Always read the questions carefully for words like "not" or "except" and read between the lines because the questions can be really tricky. Know the difference between "imply" and "infer" because they're very common in questions. If you're not sure of an answer, just choose the one you think is best, then flag it and come back if you have extra time, but don't leave any question unanswered hoping to come back to it.
Read the news for current affairs and try to be selective with what you read. Read sample essays and practise writing yours. If you aren't already fast at typing, try and improve your skills before the day.
Goodluck!