The FedSuns didn't surrender and have the government dissolved. Yori Kurita didn't stand over the first prince's corpse, pillow in hand, declaring there is no more FedSuns. thanks to an unconditional surrender and dissolution. This isnt a good comparison.
Your sequencing is backwards and that messes up your conclusions. RAF troops didn't flee the Republic before it died. The remains of their homeland aren't undefended. The conquest is already done. Every single planet save Northwind was conquered by neighboring states. There's no Republic left to defend. None of the RAF troops have been shown to have gone merc until after their government was disbanded, which means none have been shown breaking their oaths to the Republic. This idea that people would think they are deserting oathbreakers isn't based in what actually happened in setting.
It also doesn't follow setting precedent. There's a much better comparison than twisting the fall of New Avalon as a comparison. The Star League. Multiple merc units - including some of the most famous and notable ones - started under exactly the same circumstances (dissolution and collapse of the Star League / Hegemony) and nobody is concerned. If anything, the desire to uphold ideals of a better age is a selling point.
That doesn't even get into the realpolitik of the moment in the setting: Machines are plentiful, pilots are not. People are desperate for well training soldiers and there's a new supply.
There's no reason to expect Galatea or other hiring worlds would be averse to ex-RAF mercs. Their main concern would be stolen valor, where people pretend to be ex-RAF.
The star league was a freak in two ways that the Republic isn't.
1. There were still member states claiming legitimacy. This is not the case in the Republic, assuming you accept that having your government occupied after a surrender and nation partitioned (there were still worlds unconquered by the Wolves when Terra fell) by an illegal and unlawful dissolution by an occupier of your capital. In THAT situation (the Star League), the units 'going 'merc' were still viewed as supporting their nation-either because of a majority of the troops being FROM that successor state, or because the Successor State Lord was a legitimate claimant to the Star League throne and leadership. They were not, after all, the "SECESSION" wars...
2. The real comparison to the Republic's dissolution would be the Amaris Coup, not Operation:Exodus. A foreign Army has occupied the Capital-for the second or third time, I grant, but a foreign occupier also 'dissolved' the Republic (orders given under duress are by definition, illegal and unlawful, Stone was under duress.) This makes his order to dissolve the Republic unlawful, obedience of that order, therefore, is desertion in the face of an enemy. Soldiers are bound by their duty, and the Republic was, at least on the face of it, a nation comprised of member worlds-those member worlds didn't immediately declare independence or bow to the invaders, therefore, the RAF units that went mercenary betrayed their oaths, and revealed themselves to be faithless.
To put it further, Terra is fallen, the supreme leader surrendered, and then died-conveniently timed just after the surrender. These SOLDIERS are still tooling around in billions upon billions of [currency] of public property that they do not, in fact, legally own. We can add theft of public property, while abandoning worlds that had not yet surrendered or even yet acknowledged said surrender, which is desertion and dereliction of duty.
Their oaths are, therefore, absolute shit. worthless, they cannot be trusted to uphold their contracts, since they did not leave those contracts in a legitimate fashion, they are in possession of military grade hardware with no legitimate reason to have it.
They are, by common definition, brigands, bandits, and possible Pirates.
