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The easiest tents to set up are freestanding tents.  These generally require the fly vestibules to be staked out but otherwise are self supporting.  For 2 person backpacking there are light weight options like the MSR Hubba Hubba or  Big Agnes Copper Spur 2.  but at over 3# they are now considered on the heavier side for solo carry. 

The next lighter are semi free standing tents like the Big Agnes Tiger Wall 2 and REI Quarter Dome 2 SL which generally have a three "feet" instead of four and require the  foot end to be stakes out in addition to the vestibules. I use a Tiger Wall 2 for solo carry.

Apart from stretching your budget,  a problem for you with this kind of tent as they get lighter the fabrics get thinner and a dogs claws are likely to rip the floors.   If you get this kind of tent you might want to make a liner for the inside floor out of tyvek ...this is often used to make in expensive ground cloths/footprints to use under the tent to protect it.

If you are not hiking far then an alternative is to get a heavier tent like the REI Half dome 2 which is made of 40D cloth and fits better with your budget.  REI carry a number  of budget tents in the 4 to 5 lb weight range that might withstand a dog better.

If you do intend to hike a bit further and/or weight is and issue and you carry a trekking pole you might consider a trekking pole tent.  These tents save weight by using using only one or two straight poles or trekking poles.  They are not free standing and have to be staked out so you need a bit more skill to erect them and if you normally camp on rock you will have to use rocks or trees to guy to.  Usually they are single walled you are not protected from any condensation that builds up on the fly.  Carry a microfiber towel and wipe it off before it drips on you.  However they do save you some weight.   One to consider in particular to share with a dog is the well regarded  Six Moons Designs Luna solo which is large enough for 1.5 people and is within your budget.   It uses  40D fabric for the floor which should hold up better to dog claws.  I may still try to protect it though.  If you order one you should probably add seam sealing and if you don't carry a trekking pole you can order a single tent pole to use instead.  Unfortunately REI does not carry this tent.  If you are tempted I would definitely watch some YouTube reviews so you understand what you are signing up for.

Note:  REI have just this year released the Flash Air trekking pole tents and they are currently on sale.  I am tempted to try a Flash Air 1.  However for your use, the 1 is probably too small and the Flash Air 2 requires 2 trekking poles which may not be practical with a dog unless you carry the provided poles instead which adds weight and is probably a bit more of a pain to set up negating its advantage.   I also think the floor fabric on these tents is quite thin (I think I read it was 15D but I may be wrong)

Lastly for trekking with a dog you might checkout the YouTube channel, HomemadeWanderlust (aka Dixie) as she took her dog ( and her mother ) on a fill in trip along the PCT last year