Kevin that is exactly what I am getting at.    Depending on if the weld is conduced from the outside surface or from the inside of the bore will determine your options and approach.    Is this hole simply filled with weld or is an actual plug put in place and welded out?    If its just filled with weld...maybe using an actual plug and then welding should be something to consider if design constraints would allow it.

Dave offers a sensible straightforward answer :  If you have tight tolerances to hold saving a clean up/finish bore for after the weld could be the very best recourse you have.    If thats not possible at all below is my $.02 on what you can do from the welding side.

Both copper and aluminum are great for heat sinks...both metals conduct heat rapidly.   A bar of copper turned to the bore diameter and then sliced lengthwise to a little less then half the diameter could do much to help dissipate the heat and egg shaping that bore.  This in conjunction with a piece of  thick bar stock  on the outside flat wall with a hole drilled in it big enough for welding access...possibly/probably could solve your issues.   Whatever you use it is crucial that your sink makes solid contact with as much surface area as possible and is somehow clamped to that surface.  That wall is does not look that thick (in relation to bore dia) and I do not know what tolerances you need to achieve but with the right setup you should be able to eliminate the distortion at all but a small area right at the weld site.  Water cooling/Air cooling your heat sink is where I get the term "chiller block".  It is not hard to fabricate one that uses a gravity fed supply of water or you can go all out and use a circulating pump and tank.  All you need is enough flow to prevent boiling of your water/coolant.  Air cooling with many small holes drilled thru the heat sink all interconnecting with compressed air applied can do much as well.   More extreme measures involve actually icing down your heat sinks..but this can introduce cracking and other problems so I do not recommend that.

One last option I can offer would be to weld the hole shut in stages, allowing it to cool to about 200 degrees in between passes, using much smaller filler about 1/2 the amperage.  This would increase your welding time greatly and probably by itself would not eliminate your problem, but likely to reduce it quite a bit.

Someone might have some commentary about preheating and uniformity of temperature but I don't know any techniques in that area that I believe could help.

Keep us informed of your progress with solutions.  Good luck.

Best Regards
Tommy