Some of the problems at Basingstoke could be resolved by putting a dive under from where the current long valley siding is to pass under the mainline to emerge at the London end of Barton Mill sidings on the up side to join the US line there.
Just as the current bay platform for Reading line trains could be extended to Winklebury before either doing a flyover/dive under or via a set of fast crossovers to join the mainline there.
This would help remove the conflicting and congesting moves at the station.
The potential problem with dive unders is that there are a number of roads and footways which pass under the railway line. Even looping a line through and or behind platform 5 could be difficult because of the already fairly low road bridge at the western end of the platforms which will only get lower if the bridge is extended North as the road is going up at this point. Which would mean that you would probably need to raise the level of the track.
Although electrification to Salisbury wouldn't be cheap it could be cheaper and/or easier and/or less disruptive than changing the junction at Basingstoke. Although personally I think that it would be best to do that, so as to remove the Basingstoke stoppers from crossing at Basingstoke on the flat, and then do the junction works almost straight after/as the latter part of the save project.
Just so that if you have limitations at Basingstoke due to the junction works you've freed up platform capacity in advance.
Of course the fact that it improves capacity and frequency of services to Salisbury, makes getting bimodal trains for the WofE line more likely and brings other benefits is all an added bonus.
I'm not sure if the £1.7 million cost per km on the GWML was single line costs or route km costs. However at about 64km between Basingstoke and Salisbury that would be £110 to £220 million based on those costs.
In reality it should be a lot lower than that as there would be fewer junctions, (if route miles) always only two tracks rather than sections of four tracks, less trains to cancel (and therefore less compensation to pay SWR) and many of the signals have been replaced fairly recently so the location of the equipment is much better known (so less likely to break signal cables).
That may sound a lot to remove just two junction movements an hour, but it is more the fact that the Basingstoke stoppers block platform 1 for about 5-10 minutes on the way in (arriving, making sure everyone is off before running to the siding) and about 10-15 minutes on the way back through. By running to Salisbury they would only block a platform for 3 minutes (arrive, standard station stop, leave).
Although you could gain something similar by building a new station at Oakley, electrifying between the junction and there, installing points to allow a train to turn back and maybe a siding to shunt the unit or if the way of WofE services. Although cheaper it wouldn't bring the same level of benefits (i.e. 4tph between Basingstoke and Salisbury with the potential to skip some of the minor stations from the existing WofE services)