I'm afraid 150K won't get you anything in harborne/moseley/KH. Unlikely to find much in Selly Oak either. East of Moseley is very busy/vibrant, but you will have no chance of finding a school place, as the schools are hopelessly over-subscribed I'm afraid.
You need to be near a station really, and South birmingham (moseley/KH) isn't served by the train! Nor is Harborne.
Cotteridge and Stirchley are closer to trainmen, though he'd have to go up into New St first, and out again.
This is in Harborne (and slightly over budget). It is a very naice road, and close to Chad Vale Primary school, which is a good school if they have any spaces. It's the kind of area that professionals do move to/from for work though. It would be simple for DH to get the bus into city centre to station, however, it can take quite a while when it's busy (i.e. any time after 7:55am)
This one is Harborne/Edgbaston/Selly Oak border which is a bit of a difficult to classify area. It would be a short walk up to the University station for DH, and he would need to go into New St (5mins/3 stops) to change for London trains. Schools available would be St Peters (at a push, would depend on waiting list) Watermill (nice, one form entry, friendly school. High turnover rate as they take a lot of children whose parents work at the University or at the hospitals, so move fairly frequently), Paganel (not so nice, though I believe improved). Lots of the houses here are rented out to nurses/young professionals, and some to students.
There are other options, closer to city centre- the old Woodview area has been completely rebuilt, and there are lots of flats to rent there (2 and 3bed). the advantages are - v close to city centre (i.e. 10 minutes walk, even for DS), close to station (either Five Ways, into New St, or straight walk into New St). There are houses too, though I didn't see any on right move at the moment. The negatives are schools- the inner city schools are not wonderful, at least not on initial glance at performance tables. They have a lot of pupils with EAL, and on pupil premium. However, most of them have good rates of progress. I do know though, that as with most areas of Bham, getting a school place will be difficult, and you may end up travelling some way for school (though transport links from there are v good, road and bus) .
This is central Kings Heath and 20 yards from an RC Primary School, 20 yards from the high street, and supermarket, banks, medical centre etc. Equidistant between three other (non-faith) primaries. Positives- great vibe in KH, reasonable schools, fantastic parks, all very close, very walkable. negatives- well parking nigh-on impossible. School places very difficult, very over-subscribed, despite schools being three form intakes (Colmore, KH Primary, Wheelers Lane (two or three, can't remember) St Dunstan's two form). Other negative is the distance DH would have to travel to get the train. It can be over an hour into/out of city centre at rush hour.
This is central Bournville which is a v very nice area. 15 minute walk to station for DH (because you have to go round, cannot cut through), trains into Bham, then out to London on other line. Close to Bournville Infant School which is good, but will always have a waiting list. There's also Raddlebarn and St Edwards (RC) that are walkable (though longish walk) and easier to get into, but still likely to have waiting lists in Y1.
This is in Cotteridge which is popular and less expensive than B'ville. There's a school a short walk away, and one a little further (Stirchley) though neither is as sought-after as Bournville Infants. It's right next to a lovely park, and an easy walk to train station (though again, would have to go into city centre, and out again for london). There are lots of affordable options in this area though, and lots of MNers live in the environs. 
Birmingham is weird though. There are some areas that seem perfect for families, with affordable housing... and then you find out you cannot get a school place anywhere, not because of over-subscribed-ness, but because there's no school 
. Other areas there are four or five schools within spitting distance of one another (and you can't get a place because they're all full!)
Happy to help if you want to know about specific streets/areas/schools etc. If you put a thread into chat, lots of Bham MNers will come out of the woodwork- they always seem very secretive, but very enthusiastic about recommending their patch when people ask! 