Honestly, as a TA who works in Y3 at the moment, she doesn't sound that far behind. She's definitely ahead of lots of the lower attaining children in my class. New research this week has shown that the current year threes are the ones hardest hit by the pandemic - their last full year at school was reception!
I think you're right in thinking a tutor would be too much for her at this age. I would focus on the reading (the spelling will come - I'm always telling children the best way to get better at spelling is to read lots as then you start to see the words spelt correctly. I was a terrible speller at that age but now as an adult I can generally tell if a word is spelt wrong because it "looks wrong". I'm sure this has come from years of reading and seeing words spelt correctly) and the maths.
Keep reading with her lots (and reading harder books to her to help with vocabulary and comprehension). Don't forget to keep stopping and asking questions from simple retrieval questions where the answer is right there in the text, to more complex ones like "how is xxx character feeling right now? How do you know?" and "What do you think might happen next?". If she's struggling with a word, resist the urge to read it for her. Encourage her to use her phonics, remind her of alternative sounds a letter might make (things like g making a hard g or a soft j sound, c making a hard c or a soft s sound). If it's a long word, help her split it up into syllables by covering up parts of the word to just show a syllable at a time, or if she can say all the sounds but struggles to blend it together into a word, say the sounds again for her faster so she can hear them and hopefully get the word.
Lime is only one band below Brown, which is where she should be by the end of Y3.
With maths, do your school have Times Table Rockstars? If not, it's worth looking into (they have a family subscription but I don't know how much it costs). We've found this amazing with some of our children as the repetative nature is just what you need with times tables. The children want to earn coins to spend in the shop and try and beat their scores. They recently introduced a new "jamming" mode where you can choose which times table(s) to practise and it's not timed, which can reduce the pressure.
We tell our class that they should be able to answer times table questions as quickly as they can answer "what's your name?" I would go right back to the ten times table and work on instant recall by practising, practising, practising. On the way to school, over breakfast, etc, whenever you have a minute.
Same with mental addition and subraction. Work on strategies - if I'm adding a two digit number to a two digit number, counting on is unlikely to be the best strategy, but I could add the tens together, add the ones together, and then combine. Start with jotting down each mini answer, then move to doing in completely in her head. For example, to do 34+29, I would add the tens, so 30+20=50, then the ones: 4+9=13, then combine 50+13=63.
I think at this age, little and often is better than longer sessions with a tutor.