The bonus proficiency is a domain feature. Domains are a class feature. The multiclassing rules state that you gain the class features when you get a new level in the new class. It then lists exceptions to this with channel divinity, extra attack, spell casting and unarmored defense. Domains are not listed, so it seems that the bonus proficiency is the more specific rule and you gain the domain proficienices.

First, evasion isn’t a feat

This is pretty important, because you seem to be treating all class features as feats, and they are not. Some class features grant bonus feats, but most are not feats—which means you couldn’t choose to take them on your own outside of the class. That’s important.

Evasion is a class feature, found on a few classes like the monk and rogue that you mention. It’s also granted, for example, by a ring of evasion. But you can’t just hit 3rd level as an investigator and decide you want to take evasion as your 3rd-level feat. So already there is a mistake made with your character. Talk to your GM about fixing that, and then you can get evasion from monk.

Second, Improved Unarmed Strike is a feat

Just so we’re clear, the other “feat” you mention—that you call unarmed strike—is most likely actually a feat, specifically the feat Improved Unarmed Strike (all creatures can make unarmed strikes by default, no feat required; the feat just improves that option). Rogues don’t usually get that as a bonus feat, but all characters are allowed to freely choose a feat at 1st—that is probably what you meant.

Monks do get Improved Unarmed Strike as a bonus feat. Awkwardly, the name of the class feature that gives it to them is just “unarmed strike.” This is different from the default attack option that all creatures can do—it’s a particular class feature that grants the Improved Unarmed Strike feat as a bonus feat, and then some on top of that.

Third, your actual question

If you multiclass and the new class grants a feat you already possess can you choose to pick another feat that stacks with the existing one?

The short answers is “it depends, but not for the examples you chose.”

Generally speaking, you have to look to a particular feat or class feature to see what it does and how it works in conjunction with separate copies of itself. If it doesn’t say anything, then getting it a second time does nothing for you. There’s an FAQ confirming that for class features, and the rules for feats have always been that way (and my preference, d20PFSRD.com, seems to be missing this information, so my thanks to ShadowKras for the PRD link).

But some feats and class features do have particular benefits when you get it twice. For example, compare evasion:

At 2nd level and higher, a rogue can avoid even magical and unusual attacks with great agility. If she makes a successful Reflex saving throw against an attack that normally deals half damage on a successful save, she instead takes no damage. Evasion can be used only if the rogue is wearing light armor or no armor. A helpless rogue does not gain the benefit of evasion.

with uncanny dodge:

Starting at 4th level, a rogue can react to danger before her senses would normally allow her to do so. She cannot be caught flat-footed, nor does she lose her Dex bonus to AC if the attacker is invisible. She still loses her Dexterity bonus to AC if immobilized. A rogue with this ability can still lose her Dexterity bonus to AC if an opponent successfully uses the feint action (see Combat) against her.

If a rogue already has uncanny dodge from a different class, she automatically gains improved uncanny dodge (see below) instead.

^{(emphasis mine)}

See how uncanny dodge, unlike evasion, specifically handles the situation where you already have it? It grants improved uncanny dodge instead. Evasion doesn’t do anything like that. So if you did have evasion (and again, as an investigator, you shouldn’t), taking your 2nd-level of monk and gaining evasion again wouldn’t do you any good.

Likewise, when we look at the Improved Unarmed Strike feat, we have this:

You are skilled at fighting while unarmed.

Benefit: You are considered to be armed even when unarmed—you do not provoke attacks of opportunity when you attack foes while unarmed. Your unarmed strikes can deal lethal or nonlethal damage, at your choice.

Normal: Without this feat, you are considered unarmed when attacking with an unarmed strike, and you can deal only nonlethal damage with such an attack.

Compare that to the Weapon Focus feat:

Choose one type of weapon. You can also choose unarmed strike or grapple (or ray, if you are a spellcaster) as your weapon for the purposes of this feat.

Prerequisites: Proficiency with selected weapon, base attack bonus +1.

Benefit: You gain a +1 bonus on all attack rolls you make using the selected weapon.

Special: You can gain this feat multiple times. Its effects do not stack. Each time you take the feat, it applies to a new type of weapon.

You see that Special section? It says you can select the feat multiple times, and tells you what happens when you do. A feat that doesn’t have that, like Improved Unarmed Strike, cannot be selected again once you have it, and if a class or other effect grants it as a bonus feat, it does you no good.

Fourth, what you can do about it

So do I just eat those bonus feats (i.e. I already have them so tough luck)?

Well, you can just do that. If you don’t do anything to avoid that fate, that is where you would end up. But you have options here.

Houserules

The simplest thing, from a player’s perspective—the GM can change the rules of the game to avoid these class features going to waste. The GM can decide to change evasion to grant improved evasion if you get it twice, the way uncanny dodge does with improved uncanny dodge. The GM can institute a global rule that if you receive a feat you already have as a bonus feat, you get to choose some other feat you qualify for, instead. These kinds of houserules are pretty common, but as a player you can only ask your GM about them, whether or not to actually implement them is up to him or her.

Retraining

Many games allow some form of retraining, that is, changing the choices you made at previous levels. Paizo even published their own optional rules for it, which are used by Pathfinder Society games. Other games may use those rules, or use other rules that allow similar things; ask your GM.

The idea, then, is to retrain feats you are getting elsewhere to another option. This doesn’t help with your doubled evasion—neither monk nor rogue gave you a choice about that so you can’t change anything about it—but it could allow you to choose something other than Improved Unarmed Strike for your 1st-level feat, since monk is giving it to you as a bonus feat anyway.

Archetypes

Every class in Pathfinder also includes a number of archetypes, that is, variations on the class that trade some class features for another. For example, several monk archetypes trade away evasion, so if you already had it (which, again, you shouldn’t as an investigator), you could take one of these archetypes and avoid getting a redundant evasion ability.

• Monk of the iron mountain gets Toughness as a bonus feat instead of evasion.

• Sensei gets insightful strike instead of evasion and the 2nd-level bonus feat.

• Terracotta monk gets trap intuition instead of evasion.

• Weapon adept gets Weapon Focus, and later Weapon Specialization, as bonus feats instead of evasion.

• Zen archer also gets Weapon Focus, and later Weapon Specialization, as bonus feats instead of evasion, but must choose a bow for those feats.

Note that each of these also trades away other things, and you can’t pick and choose, so read each carefully.

Unfortunately, no monk archetypes replace Improved Unarmed Strike, so you’ll have to use retraining to change your 1st-level feat if you don’t want to waste that. You also could retrain your existing class to a new archetype, to lose the original evasion, if you had levels in a class that granted evasion (again, you don’t), and that way avoid having to take one of the above monk archetypes or get a redundant evasion ability.