Ok, well I can see why you're struggling- there's just not much of that size available at present!
This is ok, you'd get into Kings Heath Primary from here. Don't know how long it's been on the market I'm afraid. Or this, which is on a really, really nice road, again KH primary catchment.
I think 3 beds would be a better bet, particularly if you wanted Colmore (more academically orientated than KHP).
If you wanted Moseley CE (usually rated best primary in Birmingham) then you have to have a letter from the vicar confirming two years attendance at CE church, ordinarily St Mary's in Moseley, but St Anne's is OK, and I have known them accept All Saints KH previously. If you don't have that, you need to live within about 100 yards, though I think at present all places are going to siblings and church-goers. (houses that close at present are all £725k+)
KHP is a bit more hippyish than Colmore.
Woodthorpe and Wheelers Lane are ok too though, but Wheelers Lane, Colmore, and KHP are all v big- 3 or 4 form intake. Woodthorpe and Moseley are smaller.
You don't want anything that will give you Park Hill primary or Anderson Park primary, or Moor Green.
Both KH and Moseley will give you that vibrant cafe, shops, restaurants, bars experience tbh. KH is better on parks, however, Moseley private park is gorgeous- you pay an annual subscription to get a key, but you need to live in area.
I will say that getting up to Derby a couple of times a week is not going to be fun. It's about an hour and 10 mins when there's no traffic. I dread to think about times when the roads are busy- sometimes KH to Bham city centre can take 50 minutes at rush hour. There are no trains in either Moseley or KH.
Perhaps North Birmingham/Sutton Coldfield would be a better option? Somewhere like Four Oaks, for example?